Poehling took a shot from just above the goal line that went off the skate of Oilers defenseman Darnell Nurse and was nudged just across the goal line by the skate of goalie Tristan Jarry, a video review confirmed.
The Ducks have a 3-1 lead in the best-of-7 series heading to Game 5 on Tuesday in Edmonton.
Cutter Gauthier, Mikael Granlund and Jeffrey Viel scored, Jackson LaCombe and John Carlson each had two assists, and Lukas Dostal made 24 saves for the Ducks, who have won three in a row.
Evan Bouchard had a goal and an assist, Kasperi Kapanen and Ryan Nugent-Hopkins also scored, and Connor McDavid had two assists for the Oilers, who had won 10 straight Game 4s dating to 2023.
Jarry made 34 saves for Edmonton in his first Stanley Cup Playoff start in four years.
Kapanen scored his fourth goal of the series and the Oilers scored first for the fourth straight game to take a 1-0 lead 38 seconds into the first period. Jake Walman took a shot from above the left circle that was blocked by Beckett Sennecke. Mintyukov then failed to clear the puck from in front of his crease and Kapanen was there to shoot it into the net.
The Oilers were on the first power play of the game when Nugent-Hopkins took a shot from the left face-off circle that hit the pad of Dostal, came out sideways, banked off the skate of Mintyukov and into his own net to give the Oilers a 2-0 lead at 6:32.
Josh Samanski was called for tripping Drew Helleson at 6:13 of the second and Gauthier scored up high on the short side from the right circle on the ensuing power play to cut it to 2-1 at 7:36 of the second period.
The Ducks went back on the man-advantage at 17:00 and Granlund scored from the right hash marks off a give-and-go with Leo Carlsson to tie it 2-2 at 18:43.
Edmonton scored its second power-play goal of the game four seconds after it began on a snap shot by Bouchard from the right circle, moving the Oilers back in front 3-2 at 3:27 of the third period.
Viel scored for the second straight game after Carlson’s one-timer from above the left circle hit him in front of the crease and he swept the loose puck into the net as he fell to the ice to tie it 3-3 at 13:31.