Pouliot kept the Oil Kings at bay in the third period, turning aside all 19 he faced in the third period en route to his fifth win of the season.
Mathew Dumba had a pair of goals for the Rebels (22-13-3), while Wyatt Johnson, Turner Elson, and Brooks Maxwell had other markers.
Michael St. Croix, Travis Ewanyk and TJ Foster scored for Edmonton (27-8-5), who saw their winning streak snapped at five games.
Johnson opened up the scoring for Red Deer just 1:18 into the contest, streaking in on the left wing and chipping the puck over the right pad of Oil King netminder Tristan Jarry.
Two quick goals by Red Deer put the visiting side up 3-0 before the game was six minutes old. First, Elson jammed home a rebound behind Jarry, then Dumba's power-play blast from the point slipped through traffic.
St. Croix put Edmonton on the board at 12:17, making no mistake on a defensive miscue. Ewanyk had an easy tap-in on Henrik Samuelsson's centring feed to pull Edmonton within one at 18:36 of the first.
The Rebels restored a two-goal margin at 3:27 of the second, Maxwell flying in to flick in a rebound and make it 4-2 Red Deer.
Foster hauled Edmonton back within one at 16:18 of the second, finishing a rebound off Pouliot's left pad, before Dumba added the insurance goal for Red Deer on the power play at 16:32 of the third.
Jarry turned aside 34 shots in net for the Oil Kings.