Dogs have clearly become an avant-gardist’s best friend. First Jean-Luc Godard delivered a funny 3D valentine to a pooch named Roxy Mieville in “Goodbye to Language,” and now the New York-based musician/performance artist Laurie Anderson has woven a tide of personal stories, insights and visual-musical riffs into a more accessible but no less singular consideration of the species in “Heart of a Dog.” While this alternately goofy, serious, lyrical and beguiling cine-essay serves primarily as a loving tribute to the memory of Anderson’s rat terrier, Lolabelle, its roving, free-associative structure brings together all manner of richly eccentric musings on the evasions of memory, the limitations of language and storytelling, the strangeness of life in a post-9/11 surveillance state, and the difficulty and necessity of coming to terms with death.

Wielding a darkly playful sense of humor that cuts through any poetic preciosity, Anderson’s unexpected but entirely welcome return to filmmaking (this is her first feature since her 1986 concert doc, “Home of the Brave”) should unleash enough critical admiration to win over discerning arthouse-goers. After a prolific fall-festival run, it opens theatrically Oct. 21 through Abramorama before an early 2016 airdate on HBO. Anderson’s fans will also find some of the film’s elements on display in her concurrent installation “Habeus Corpus,” which will premiere in October at New York’s Park Avenue Armory.

A pioneering figure on the New York art scene since the 1970s and ’80s, Anderson may have mellowed a bit since her avant-garde heyday, when she made a name for herself with her innovative multimedia performances and voice-altering, genre-bending experiments in electronic music. But her more recent work, up to and including this Arte TV-commissioned “philosophy of life” project, has been no less adventurous in its embrace of creative forms, including a stint as NASA’s first artist-in-residence; vocal turns in “The Rugrats Movie” (1998) and PBS’ “American Masters” documentary on Andy Warhol (2006); a well-received 2010 album, “Homeland,” which she produced with her husband, the late musician Lou Reed; and a “Music for Dogs” concert outside the Sydney Opera House, some of it performed within an aural register detectable only by canine ears.

An indulgence of all things furry and four-legged is similarly front and center in “Heart of a Dog,” which, though apparently unrelated to Mikhail Bulgakov’s 1925 communist satire of the same title, proves no less concerned with the subtler points of connection between two species that have always regarded each other with mutual affection. The film begins with a re-created dream sequence in which Anderson describes her intimate, almost maternal sense of attachment to Lolabelle in mordantly funny terms (illustrated by the artist’s own monochrome ink drawings). It’s no accident that, near the end of the film, the director supplies a moving bookend to this opening sequence by recalling her mother’s death, and the strange, conflicted feelings of affection, estrangement and regret that it awakened.

As metaphors go, it sounds cruder on paper than it plays onscreen. Rather than drawing simplistic formulations, the film unspools a micro-budget mixture of animation, visual effects, 8mm home-movie clips, newly shot footage, and variously doctored or distressed images, all of it given shape, weight and direction by Anderson’s inimitable, ever-present voiceover and characteristically inventive use of music. It’s a style so structurally liquid and formally stimulating that it naturally invites us to forge parallels among the various sights, sounds, impressions and anecdotes she’s assembled here.

And so, Anderson’s memory of a long-ago Bay Area hiking trip — during which a circling hawk swooped down and almost mistook Lolabelle for an unusually large rabbit — turns her thoughts back to the very different airborne predators that struck on Sept. 11, 2001. It’s a subject that never seems far from the mind of this quintessentially New York artist, though she’s less interested in exploiting that day’s specific trauma than in reflecting on the innumerable changes that transpired in its wake. She remarks on the suddenly ubiquitous presence of armed troops throughout Lower Manhattan, as well as the Dept. of Homeland Security’s practice of recruiting dogs and sending them to be prison for bomb-squad and military-patrol training.

But just as you wonder if “Heart of a Dog” might be settling into an animal-rights polemic, Anderson’s focus calmly shifts yet again in a work that stubbornly, charmingly refuses to settle for an obvious topic or insight. This is a movie that likes to sniff around. A bemused reflection on the verbiage of post-9/11 unease (“If you see something, say something”) can prompt Anderson to deliver a miniature discourse on Wittgenstein and his thoughts on the unique power of words (“The limits of my language mean the limits of my world”). A mention of the limited color spectrum visible to dogs gives way to green-tinted drone-camera footage. And the director lingers with particular fascination on shots of the NSA’s headquarters in Utah (shades of “Citizenfour”), almost marveling at the U.S. government’s frighteningly unprecedented campaign of surveillance against its own citizens.

While Anderson’s work has always been steeped in social conscience, what seems to intrigue her most about all this mass data collection is its narrative potential — the sheer number of storytelling permutations available to anyone who might be inclined to splice together a coherent chain of events from emails, phone calls, social-media posts, personal-information snippets, spy-cam images and other stray Cloud particles. And so “Heart of a Dog” becomes both a demonstration and a critique of the art of storytelling, which is to say the art of making meaning from the random effluvia of daily life, in ways that can both deceive and enlighten.

In typically introspective fashion, Anderson turns her gaze back upon her own tendency to censor or misunderstand the truth: When she flashes back to a serious childhood head injury that led to an extended hospital stay, she’s creeped out by how she’s subconsciously suppressed some of the more unpleasant aspects of the experience. Another harrowing, frozen-in-time memory from the same period, depicted in fragments of hauntingly manipulated 8mm footage, provides an emotional and dramatic climax of sorts, as well as a crucial insight into our sometimes faulty perceptions of how we are regarded by those we love.

For all this, “Heart of a Dog” never becomes top-heavy or self-serious; humor has long been a defining facet of her work, and it’s especially suited to her choice of subject here. In one of the more delightful passages, Anderson describes how Lolabelle, who went blind in her later years, began taking piano lessons using a special dog-friendly keyboard; eventually she excelled enough at her art to give her own concert, with the proceeds going to benefit animals in need. No fewer than six canine performers — including Lolabelle, three more terriers, a German shepherd and a poodle — are given top billing in the cast, higher even than Anderson’s West Village neighbor Julian Schnabel (who has a blink-and-you-miss-it cameo in footage shot from a dog’s-eye perspective).

Of all the resonant bits of wisdom that Anderson leaves us with, it’s her eminently quotable suggestion that we learn to “feel sad without being sad” that best encapsulates the film’s melancholy but never maudlin spirit. Inevitably, this is a movie that concludes with an extended rumination on death and the afterlife (complete with references to the Tibetan Book of the Dead), but Anderson insists on the importance of understanding each passing as not just an occasion for grief, but as “a release of love.” Viewers may well glean that Anderson is talking about not only her mother and Lolabelle, but also Reed, who died in 2013, and to whose “magnificent spirit” the film is dedicated. His song “Turning Time Around,” which plays over the closing credits, is one of a handful of stirring musical selections here (including tracks from “Heartland” and Anderson’s 2001 album, “Life on a String”) that reverberate with personal history — a reaffirmation, as if it were needed, that Anderson’s puppy-love collage is no less a portrait of herself.