Stutzle tied it 2-2 with a short-handed goal at 18:53 on a 2-on-1 with Dylan Gambrell.
"After the first period, as a group, we thought we could be better," Foegele said. "In that second period, I thought everyone was kind of going and we engaged, and we were really on the throttle there and we were fortunate enough to get a couple of goals."
Nugent-Hopkins gave the Oilers a 3-2 lead on a power play at 5:39 of the second period after a centering pass bounced back to him.
Draisaitl made it 4-2 with his 100th point at 16:17, a one-timer from McDavid.
Nick Bjugstad made it 5-2 with 0.4 seconds left in the second. He took a pass from Mattias Ekholm in behind the defense, cut to the net and skated around Sogaard before scoring for the buzzer-beater.
"It was deflating for sure, but in saying that, we didn't come out [for the third period] that way," Smith said. "This group is not going to quit. They're not going to give in."
Stutzle cut it to 5-3 at 5:28 of the third on a one-timer from Drake Batherson.