Connolly scored after a stretch pass from Eller at 2:08 of the second for a 3-0 lead.
"We need to have a better start," Penguins coach Mike Sullivan said. "We need more pushback from the drop of the puck. Listen, we knew they were going to come out hard. We're up 1-0 (in the series) after the first game, we knew they were going to come out hard. We didn't match the urgency."
Holtby helped preserve the lead for Washington in the second period when it was outshot 16-6. He stopped Hornqvist, knocking the puck out of the crease with his stick at 10:56 and then stopped Guentzel's rebound chance with a pad save at the left post.
"I thought Braden was real solid all game," Capitals coach Barry Trotz said. "I thought he made some timely saves. He moved well. They were trying to make everything very difficult and he responded the way he always has."
Letang's point shot at 13:04 of the second made it 3-1.
The Penguins appeared to make it 3-2 at 9:03 of the third period when Hornqvist tucked a puck underneath Holtby. However, it was not ruled a goal on the ice and the call stood after a review.