Richard Panik and Conor Sheary scored for the Capitals (9-6-3), and Vitek Vanecek made 34 saves.
The teams play here again Thursday.
"We don't like losing to those guys," Washington forward Tom Wilson said. "These are big points, big games, and I think if we put a little more emphasis into the full 60-minute game, maybe we can get an upper hand and get a goal lead or something like that.
"That being said, if it goes to OT, we've got to get it done. But it's a bit of a crapshoot at that point."
Kapanen, who also had an assist, responded for the Penguins after he was benched by Sullivan in the third period of a 3-2 win against the New York Islanders on Saturday.
"I think it was a good time to have a conversation with him and kind of reset and look at the past games that I've played, which obviously haven't been good enough," Kapanen said. "Just kind of talking about what he expects out of me and to play with [Sidney Crosby] and Jake, it's obviously a privilege. I think our line played better today and I felt like I took strides forward in the right direction, but there's still room to improve."