A blast of school’s-out exuberance, a roar of youthful rage, fine melodies, a big dose of tenderness, a goofy name. That’s the Goo Goo Dolls, whose L.A. headlining debut at the Roxy on Tuesday was a bracing distillation of ‘80s heartland American independent rock.

That means that the Buffalo-based trio (which also plays the Casbah in San Diego on Saturday) sounded a lot like the Replacements and Husker Du. But the group also served notice that once it outgrows its sources, it could become a standard by which future baby bands are measured.

The Goo Goo Dolls might know hangovers and heartbreak, and they hammer out a relentless thrash on stage, but their personality is unmistakably sweet, innocent and almost painfully sincere: During a brief acoustic interlude, singer-guitarist Johnny Rzeznik prefaced his character study “James Dean” with an earnest explanation of the story line, to demonstrate that it’s not anti-gay.

When singer-bassist Robby Takac wasn’t racing around the stage like a wind-up toy and singing in an Ozzy rasp, he’d twist his face and charge the crowd like a bratty kid trying to gross out his little sister. This long-haired, chubby prole and the slightly more stylish Rzeznik (shag haircut, earrings) form a distinctive team. (Drummer George Tutuska stays in the background and does his job.)

Rzeznik is the repository of suburban soulfulness, in the tradition of the Replacements’ Paul Westerberg. He sings longing, defiant rockers with that same scratchy delivery, clenching his voice to hold something back, scraping it across the churning instrumentation like a violin bow.

The set’s non-originals ran from Elvis Presley and the Temptations to the Stones, Blue Oyster Cult and the Plimsouls. They didn’t do their version of Prince’s “Never Take the Place of Your Man,” maybe because the Buffalo lounge singer who performs it on their new album “Hold Me Up” wasn’t along. Despite their hipness and humor, the absence of real wickedness made their put-downs of arena-rock cliches and Guitar Institute graduates seem clumsy.