Norris scored to make it 5-4 at 16:18 of the third period on a wrist shot from the right face-off circle. He gave the Senators a 1-0 lead at 10:35 of the first period.
Zack Kassian tied it 1-1 on the power play at 19:12, scoring on a shot from the slot off a pass from Leon Draisaitl.
Kailer Yamamoto stumbled but recovered to score on a breakaway at 12:18 of the second period to give the Oilers a 2-1 lead. Brendan Perlini extended the lead to 3-1 at 17:34.
"That's one we let slip away, we're going into the third period with the lead, and we couldn't wrap it up," Kassian said. "We need to wrap it up, we need to win that game."
Gaudette made it 3-2 at 2:41 of the third period. He took a centering pass from Formenton and scored on a backhand. Formenton tied it 3-3 at 5:07, scoring on a rebound off a shot from Gaudette.
"It was a lot of fun [and], personally, it's been a while since I've had that much fun playing hockey," Gaudette said. "It's been a tough year and a half or so, so it really feels good to contribute and to help this team win. That's what I'm here for, to do anything possible to help this team progress and win.
"It's a great group of guys in the locker room and credit to everyone, we really stuck with it, down two goals, we didn't roll over, we just came back and played our game and stuck with our system and good things happen because of that."