Evgeny Kuznetsov, Tom Wilson and Lars Eller also scored for the Capitals, the second wild card in the East. Vitek Vanecek made 30 saves.
"We liked the game we were playing the whole night and we just felt like if we stuck to it and stuck with it and played good defense that the offensive chances would keep coming and that eventually we could crack 'em," Capitals coach Peter Laviolette said.
Sam Bennett and Claude Giroux scored for the Panthers, who earned the Presidents' Trophy for best regular-season record in the NHL. Sergei Bobrovsky made 34 saves.
Game 2 will be played here Thursday.
Oshie scored when he redirected Nicklas Backstrom's pass from the top of the left circle across the goal crease.
"It looked like it was going to be a 2-on-2 right away," Oshie said. "I was able to get a step on … I'm not sure who was over there … and you know typical 'Nicky' fashion, sauced it through a guy's legs and right to my tape. I just redirected it, didn't see it go in, so he had to come tell me that I scored. It is always nice to get that go-ahead goal in the third period and it was a great play by Nick."