Alex Chiasson scored on a 2-on-1 for a power-play goal to give the Oilers a 2-1 lead at 6:03 of the second.
"Our goal was to get out of the first period tied or better, because they're the highest-scoring team in the first (30 goals), and we did that," Senators coach D.J. Smith said. "It's on us, a couple of mistakes just on changes. Our penalty kill did a good job and there was 10 seconds left and guys were racing off, and then they had the quick-up on the other goal. Those are great shots by really good players."
Stutzle, a rookie forward, scored when his shot hit the stick of Oilers defenseman Tyson Barrie and went up over Koskinen's shoulder at 17:57 of the second period for the 3-2 final.
Three of Stutzle's six goals this season have been at Edmonton.
"I just try to play my best game every night, play hard and play my best for the team," Stutzle said. "I like the rink here, but I just try to play my best hockey every night, it doesn't depend on which team we play."
Koskinen made 11 saves in the third period for his first win since Feb. 20.
"You can't fault the effort," Smith said. "I thought the guys really tried in the third period to tie it up. But give them credit, they did a good job."