Ryan McLeod found Tuch all alone in front of the net with a pass from the left boards and Tuch scored glove side on Stuart Skinner for the game-winner.
Tage Thompson had a goal and two assists, Josh Doan scored twice, and Rasmus Dahlin had three assists for the Sabres (12-14-4), who had lost the first three games of a six-game road trip (1-3-0), including a 7-4 loss to the Calgary Flames on Monday.
Colten Ellis made six saves before he was pulled by a concussion spotter for the final 59 seconds of the first period following a collision with Edmonton forward David Tomasek, who was called for goaltender interference. He was replaced by Alex Lyon, who stopped 21 of 24 shots. Ellis did not return to the game.
Connor McDavid scored twice, and Vasily Podkolzin had a goal for the Oilers (13-11-6), who had won two straight and are 2-1-1 in a five-game homestand. Skinner made 24 saves.
The Oilers scored two goals in the opening 1:56 of the third period to get back into the game.
McDavid cut the deficit to 3-1 just 10 seconds into the third, powering his way to the net and lifting a shot over Lyon's right pad.
Podkolzin made it 3-2 at 1:56, chipping in a loose puck during a goal-mouth scramble after Leon Draisaitl sent a shot on net from the top of the right face-off circle.
McDavid tied the game 3-3 with two seconds remaining in the game. With Skinner on the bench for the extra attacker, a rebound from a Ryan Nugent-Hopkins shot slid to McDavid in the left circle, where he buried a snap shot into an open net.
Doan put the Sabres ahead 1-0 with a power-play goal at 19:01 of the first period, tipping a point shot by Dahlin under Skinner's glove.
Thompson appeared to put the Sabres up 2-0 at 9:37 of the second period, when he beat Skinner with a backhander. Edmonton challenged for a missed hand pass by Tuch, and the goal was rescinded after a review.
Thompson stole the puck from Evan Bouchard in Edmonton's defensive zone to the right of the net, skated in front and beat Skinner from in tight to make it 2-0 at 11:03.
Doan scored his second power-play goal to push the lead to 3-0 just 56 seconds later at 11:59. He took a feed from Thompson at the top of the crease and put a shot between his legs far side on Skinner.
