Beauchemin fired a slap shot past goalie Viktor Fasth, with assists from Perry and Ryan Getzlaf, as the Ducks improved to (50-18-8).
''Their team was taking away the far side, so I just tried to shoot it as hard as I could,'' said Beauchemin, who scored his 10th career winning goal. ''I knew (Perry) was going for a rebound if it was there. Fortunately, it went in. I was pretty happy with that one.''
With the Ducks trailing 2-0, Perry scored his 40th goal just 37 seconds after Jordan Eberle doubled Edmonton's lead at 2:51. Perry then tied it at 11:25 with No. 41.
It is the second time Perry has reached 40 goals in a season.
''We've been a team that's worked had all year,'' Perry said. ''We have that never-say-die attitude. If we continue to push and play the right way for a full 60 minutes, we'll be all right.''
Rookie goalie Frederik Anderson (19-5) stopped 30 shots for the Ducks, 5-0-1 in their last six games.
''That's twice he's come up with a big save,'' Ducks coach Bruce Boudreau said. ''He made a big save when it was 2-0 right after their goal. Then we score right away. What a swing.
''There's no way we would've come back from 3-0, but Freddie stood tall and made a great save.''
Philip Larsen scored his third of the season in the first period for Edmonton.
Anaheim improved to 27-7-4 at home. Edmonton dropped to 12-22-6 on the road.
''The idea of winning 50 games is pretty cool, quite frankly,'' Boudreau said. ''There's only going to be a few teams that do it, and when you're one of them, it's quite an accomplishment. I'm pretty proud of the guys for 50 wins.''
Fasth, who faced his former team for the first time since being dealt for two draft picks a month ago, made 23 saves but dropped to 2-2-1 for the Oilers.
In two seasons with Anaheim, Fasth was 17-8-3 with four shutouts and a 2.32 goals-against average.
''There were a lot of emotions,'' Fasth said. ''I had been playing with the guys on the other side for over a year. It was a special feeling. That's a tough loss.''
The Pacific Division-leading Ducks are three points ahead of second-place San Jose with a game in hand. Anaheim trails Western-Conference leading St. Louis by one point.
Edmonton, last in the Pacific Division, will finish a three-game road trip at Phoenix on Friday.
The Oilers, who have scored on just 16.4 percent of their power plays this season, took advantage of one to grab a 1-0 lead over the Ducks.
Larsen drilled the puck into the upper right corner of the net on the glove side with 3:56 left in the second period.
Edmonton outshot the Ducks 24-15 over the first two periods and took a 1-0 lead into the third.
The Oilers doubled their lead on Eberle's 26th goal of the season.
''It's heartbreaking,'' Edmonton coach Dallas Eakins said. ''It's games like these that you want your players to feel the reward of the points. When we had it 2-0, we had an unbelievable chance in front of the net and a rebound that, if it would have went to three, it would have been over for sure. But then again, with the Ducks, you never know.
''They seem to be very, very good at never being out of a game. We have to find a way to close it out.''
NOTES: Anaheim leads the NHL with 3.23 goals per game. ... The Ducks' four-goal comeback win over Winnipeg on Monday was the largest in an NHL victory since Jan. 9, 2010, when Minnesota beat Chicago 6-5 in a shootout, after trailing 5-1. ... Oilers D Oscar Klefbom scored his first NHL goal against Anaheim in an 4-3 overtime victory on March 28.