Brian Fallon, the scruffy frontman for the Gaslight Anthem, keeps a notebook in which he writes down major events in the band’s history. “It’s one sentence — like, this happened here,” he says. For example, an entry from June 2009: Bruce Springsteen joins the band onstage at the Glastonbury Festival in England. Fallon was walking up the ramp to the stage when Springsteen grabbed his shoulder and asked if he could play with them on “that ‘ ’59 Sound’ song,” the title track from the band’s breakthrough album. “I think I know the chords!” he told Fallon. It was the Jersey-rock version of being knighted. “We got offstage, ” Fallon remembers, “and I said to myself, ‘I don’t think people will view us the same after this.’ ”

The next night, Springsteen was playing an outdoor concert in Hyde Park, and he invited Fallon onstage to sing with him. “I’ve never seen Bruce with a bigger smile on his face singing with a young man,” says the rock historian Matt Pinfield, a fellow Jersey boy and Gaslight fan. “Because even Bruce knows Brian’s the heir apparent.”

Since forming in the mid-2000s, the Gaslight Anthem has emerged as everyone’s favorite rock ’n’ roll underdog. This month the band released “Handwritten,” its fourth album and first for a major label. The album is steeped in Gaslight’s signature blend of soulful punk. But it also reaches­ for the all-American classic-rock anthems that the band has been hinting at for years.

The guys in Gaslight have the easy intimacy that comes from having spent years together in an Econoline van. “Remember how much money none of us had at that point?” Fallon asked the guitarist Alex Rosamilia recently on the patio of a pub off Main Street in Red Bank, N.J. He was talking about the making of “The ’59 Sound” in 2008. They recorded it in Los Angeles while living in a complex in Burbank that also housed lots of “beautiful and too confident” child actors, recalled the drummer, Benny Horowitz. Those were the years when the band drove to the next night’s gig on the previous night’s earnings; they were on the road so much that Horowitz didn’t even bother renting an apartment. “We watched ‘Talladega Nights’ every night for two years,” Fallon recalled. “That movie was the only normal thing we had.”