Stutzle and Drake Batherson each had a goal and an assist, and Tkachuk and Thomas Chabot each had three assists for Ottawa (20-21-3), which was 4-for-9 on the power play. Cam Talbot made 15 saves.
"I thought we played a great game, wire to wire," Senators coach D.J. Smith said. "I thought our bench was under control tonight. You know, guys were dialed in to what they were doing and sometimes flukey goals go in and things like that, and then you've got to stay with it. It's easy to get down, but I thought it was a really good effort by a lot of guys."
Ottawa scored four power-play goals for the first time since Dec. 17, 2005, when it scored six in an 8-2 win against the Toronto Maple Leafs.
"It's pretty tough to assess a game like that when half of the game is special teams," Penguins coach Mike Sullivan said.
Evgeni Malkin had a goal and two assists, Sidney Crosby had three assists, and Casey DeSmith made 35 saves for Pittsburgh (22-15-7), which was outshot 40-19.
"Except for the last one, I thought I played well," DeSmith said. "The last one was not good."
Senators forward Josh Norris had two shots on goal in 19:04 of ice time in his return from a shoulder injury. He missed 38 games, last playing Oct. 22.
"He makes our team so much better with him in the lineup," Tkachuk said. "We missed him. It didn't look like he was out of shape after being out three months. He looked unreal. His speed, his skating and his shot. For me, the personal connection I have with him, I was fired up tonight."