Yakupov gave St. Louis a 1-0 lead at 3:37 of the second period on a pass in front from defenseman Joel Edmundson.
Tyler Pitlick tied it 1-1 at 6:22 when he tipped Andrej Sekera's shot from the point past Allen for his third goal. Blues coach Ken Hitchcock challenged the goal for goaltender interference, but it was upheld after a video review.
St. Louis outshot Edmonton 35-23.
"I think the shot totals were a little bit misleading, we were on the perimeter too much," Hitchcock said. "It was a 1-1 game, right where you want it on the road, and we made a big error on that 2-on-2. That gave them the lead, and we had to open it up a little bit."
Yakupov came close to tying it at 10:10 of the third, hitting the goal post on a rush after he beat Oilers defenseman Eric Gryba wide and cut to the net.
"Going into third, we were happy we responded and got the goal from [Pitlick] to tie it up, and then we would see what would come out of it," Oilers coach Todd McLellan said. "Scoring fairly early was nice and then having the ability to defend. There were some moments that were nerve-racking, but we settled things down."