Moreau received a pass from Marty Reasoner at the dot of the left faceoff circle and teed it up over the shoulder of Jose Theodore to complete the comeback. The Oilers had trailed 3-1 in the second period at one point.
Fernando Pisani iced the outcome with an empty-net goal for his second of the night.
The Oilers found themselves on the defensive early in the game after running into some penalty trouble. Michael Peca got tagged for holding 39 seconds into the game and Fernando Pisani took a hooking penalty at 2:58.
Montreal fired the first four shots of the game, but the Oilers took advantage of back-to-back power plays of their own several minutes later.
Marc-Andre Bergeron blasted a power play marker past Theodore after taking a pass from Ales Hemsky
to give the Oilers a 1-0 lead.
However, that lead was shortlived as Garth Murray responded just over a minute later with a wrister from the high slot that beat Conklin five-hole.
The Oilers ran into some penalty trouble once again late in the first after Ryan Smyth
and Chris Pronger took high-sticking penalties six seconds apart to put Montreal on an extended two-man advantage.
It only took Montreal 17 seconds from that point to score and take the lead. Mike Ribeiro fed it cross-ice to Michael Ryder who had a wide open net, taking his time before scoring the power play marker to make it 2-1.
With Pronger still in the box, Ryan Smyth
almost scored a shorthanded goal to tie it up. Smyth broke into the Montreal zone, cut in front of the net and looked to have Theodore beat but couldn't get a handle on the puck at the last minute.
After one period, shots on goal were 13-9 in favour of the visitors.
Early in the second, Montreal struck for their third of the night. Chris Higgins won a battle with Igor Ulanov in the corner of the Edmonton zone, then skated in behind the net and scored on a wraparound to make it a 3-1 game.
The Oilers came right back down the ice and scored to make it 3-2 1:01 later. Todd Harvey fed Raffi Torres for his 13th of the season.
Ethan Moreau had a good shorthanded chance midway through the second as he was able to break in alone on Theodore but he was denied.
Late in the period, Theodore looked to be in considerable pain as the Oilers concluded a power play. He was favouring his left knee and had to be attended to by Montreal trainer Graham Rynbend. Habs backup netminder Cristobal Huet came out to stretch in anticipation of coming into replace Theodore but he would stay in the net.
Edmonton outshot Montreal 11-4 in the second period but still trailed by a goal, 3-2, heading into the intermission.
Edmonton had a four-minute man advantage starting 20 seconds into the third period after Mathieu Dandenault took a double minor for high sticking.
On the ensuing power play, Marty Reasoner rang one off the post stick side behind Theodore for the Oilers' best scoring chance in the four minutes.
The Oilers tied the game up 6:05 into the third. Fernando Pisani was planted in the Montreal crease where he took a pass in front by Moreau and put it past Theodore.
Ty Conklin got his second consecutive start for the Oilers... Radek Dvorak reaggravated his groin injury and did not play tonight. Krys Kolanos suited up in his spot... Ryder moved into a tie for fourth in the league in power play goals with his ninth. He also leads the Canadiens with 14 goals... Raffi Torres tied Ryan Smyth
for the Oilers lead in goals with his 13th of the season... Pisani has four goals in his last five games... Ethan Moreau is on a career high five-game scoring streak and three-game assist streak...