The Senators led 3-0 in the second period and appeared to go up 4-1 in the frame, but lost a goal to video replay when it was determined the puck crossed the line after the net was knocked off its moorings.
Matt Cooke scored his second of the game with 7:36 left in regulation to tie it at 3. Bill Guerin had a power-play goal earlier in the third, a period in which Pittsburgh held an 18-4 shots advantage.
With Crosby held off the score sheet and Malkin limited to one assist, the Penguins got their offense from the rest of the lineup.
"It says a lot," said Crosby, who recorded 14 points through the first five games. "We're going to do our best to create things, but there are going to be nights where it's tougher than others.
"Some nights it may not go in, or whatever the case is, but we've still got to do things out there. We've still got to be productive, whether it's not on the score sheet. We've got to be responsible out there and find ways to help, but certainly there are other guys who can step up and score those goals like we saw tonight."
Matt Cullen and Daniel Alfredsson each had a goal and an assist for Ottawa, which won Game 5 at Pittsburgh in triple overtime to stave off elimination. The Senators hoped to force a decisive Game 7 back in the Steel City.
"Jordan made an unbelievable play there in the corner, beat one guy," Dupuis said. "I thought he was going to give it to me right away. He's strong enough that he decided to beat another one by himself and he slid it to me there. I'm still not sure where the puck went. I haven't seen it yet, but it's in. This is an unbelievable feeling."
Marc-Andre Fleury stopped 28 shots for the Penguins, who won for the eighth time in nine playoff series after they were eliminated by Ottawa in five games in the first round in 2007.
Staal, who also assisted on Cooke's first goal midway through the second, said the Penguins struggled to find the killer instinct in the series.
"We need to find that better and really finish them when they're down. We didn't do that," he said. "But I just love the character in this room and the way we can bounce back from things and forget about things and work for the next shift and the next game."
The fourth-seeded Penguins will have to wait for the remaining two Eastern Conference first-round series to end to know who they will face next.
Leclaire made 38 saves, and Chris Neil also scored for the Senators, who missed the playoffs last season to end a run of 11 straight postseason appearances.
"They really had us on our heels, especially in the third," said Alfredsson, who said he played despite a stomach injury that occurred at the end of the regular season. "When they scored that power-play goal we just couldn't get the momentum back. We made a lot of mistakes to lose that momentum and that's why we're done."
Leclaire, who took the loss in Game 4 after relieving Brian Elliott, gave Ottawa a strong effort for a second straight game after he was barely used down the stretch in the regular season.
Guerin drove a one-timer from the left circle off Leclaire's left arm and into the net 7:03 into the third to draw the Penguins within 3-2. Cooke tied it with a backhand into an open side at 12:24.
"We went after them, and even in overtime I thought we had a lot better mindset than we did at home, especially the first two overtimes," Crosby said. "Last game we felt like it was 50-50. We didn't really go after them. I think it was just more of a carry-over of our third period. We just tried to bring the exact same game into overtime."
Leclaire made his first career playoff start Thursday, setting a team record with 56 saves in the Senators' 4-3 win. Matt Carkner scored the winning goal in that one to end the longest game in Senators history.
Neil and Alfredsson had goals in the second period on Saturday to put Ottawa up 3-0 after Cullen scored on a breakaway in the first. Cooke made it 3-1 when he put a backhand of Staal's rebound past Leclaire at 10:56.
Cullen scored his third goal on a breakaway 5:19 in after he got open behind defensemen Kris Letang and Mark Eaton to take Alfredsson's pass up the middle. Fleury went for the poke check, and Cullen fired the puck into the top of the net to put Ottawa up 1-0.
Mike Rupp came close to drawing Pittsburgh even at 12:40. Leclaire's pad save on the goal line was subjected to a lengthy video review that supported referee Wes McCauley's ruling that the puck didn't go over the line.
Neil made it 2-0 with his third goal 1:51 into the second, and Alfredsson put the Senators ahead by three when he moved to his forehand to beat Leclaire inside the right post for his second of the series.
A very long video review overturned an apparent Ottawa goal by Mike Fisher at 16:19 that would have restored the Senators' three-goal lead.
Referee Stephen Walkom had a gash on his left cheek after he was struck by Crosby's stick late in the first. ... The defending Stanley Cup champion Penguins' six-game loss to Detroit in the 2008 finals was Pittsburgh's only defeat in the eight previous series. ... The Penguins are now assured of playing at least two more games at Mellon Arena -- their longtime home. The Penguins will move into the new Consol Energy Center next season.