Adam Erne had a goal and an assist and Robby Fabbri also scored for Detroit (4-2-1), which trailed 2-0 in the second period. Thomas Greiss made 26 saves.
Larkin beat Vitek Vanecek with a wrist shot from the left circle for the game-winner.
"It shows a lot of character," Erne said of the comeback. "[That's] a highly-skilled team with a lot of big names, and we just stuck with it, stuck to the game plan and eventually it paid off."
Alex Ovechkin scored to extend his season-opening point streak to seven games (eight goals, five assists) for Washington (4-0-3). Evgeny Kuznetsov scored, and Vanecek made 22 saves.
Washington extended its season-opening point streak to seven games but fell to 0-3 in overtime, all at home.
"We have to do a better job at managing the puck [in overtime]. The whole thing is about puck possession," Capitals coach Peter Laviolette said. "It ends in an odd-man rush usually. This was not transition [goal] since we had people back, but nonetheless it is not getting done."
Ovechkin gave the Capitals a 1-0 lead with a power-play goal at 19:05 of the first period. Greiss made a pad save on his attempt from the right of the crease but Ovechkin, down on one knee, controlled the rebound and lifted the puck over Greiss.
"It's unbelievable. I think he just gets hungrier and hungrier, honestly," linemate Tom Wilson said of Ovechkin. "This year he just seems hungrier on the net, he's shooting the puck a ton, he's getting good looks and making the most of them. When he's playing like that you just want to get it over to him and he's finding the back of the net."
Kuznetsov made it 2-0 at 4:00 of the second period when got to a loose puck after a faceoff and shot between Greiss's pads from the right circle at the start of a power play for his fifth goal.