“Sweeney” is not always easy to watch, but you can’t turn away. From our first glimpse of Johnny Depp’s haunted, vengeful eyes as the ex-convict barber sails into London on a mission to kill the man who stole his wife and child and sent him off to prison, we’re swept into Burton’s pitch-black vision. It meshes perfectly with the bitter curl of Sondheim’s lyrics. Burton isn’t interested in class warfare or the industrial-age political metaphors that Prince emphasized onstage: his “Sweeney,” from John Logan’s adroit screenplay, is an up-close-and-personal tale of blighted passions and dark obsessions. It has an emotional intimacy that allows you to see the work with fresh eyes. The object of Todd’s rage is the evil Judge Turpin (Alan Rickman), a villain right out of a 19th-century melodrama. Todd learns that while he was serving his time, his wife poisoned herself, and his now grown daughter, Johanna (Jayne Wisener), became Turpin’s captive ward, whom he aims to marry. There’s a romantic subplot in which Todd’s traveling companion, the young sailor Anthony (Jamie Campbell Bower), falls in love with Johanna, which remains the lamest, most conventional aspect of the story. The pretty Wisener and prettier Bower are decorative, and not much else.

The rest of the cast, however, could hardly be better. No one will claim this is the most beautifully sung “Sweeney Todd” he’s ever heard, but that’s not the point: they act the songs with hair-raising conviction. This anti-hero may have a one-track mind—he can’t see anything beyond the rage that drives him—but Depp is such a soulful presence he gives you a glimpse of this maniac’s pain and pathos. Bonham Carter is extraordinary. She reinvents Mrs. Lovett from the inside out—underneath her tough, practical hussy lurks a haunting melancholy, the yearning of a woman in love with an unreachable man, sucked into his madness in the vain hope he’ll open his black heart to her. Sweeney’s idea of a love song, however, is the inspired “My Friend,” a passionate serenade to his barber’s blades—he’s like a twisted Edward Scissorhands in love with his own metallic fingers. Sacha Baron Cohen (a.k.a. Borat) makes a delicious, scene-stealing appearance as the charlatan barber Pirelli, who makes the fatal mistake of trying to blackmail his rival.

The performances are pitched at just the right scale: theatrically exaggerated but untainted by “Broadway” bombast. Who knew Burton would have such an uncanny feel for how to film a musical? His strength has never been storytelling—his Gothic imagination flowers episodically—but “Sweeney Todd,” his best movie since “Ed Wood,” keeps its eye on its dark prize. It has the relentless forward momentum of a shark in blood-stained waters.