Adelaide looked in control after taking a 1-0 lead into the break courtesy of a superb strike from Sergio Cirio.
But the contest changed on the hour mark as Nigel Boogaard was sent off for brining down Bridge in the area and Tomi Juric pulled level from the spot.
Bridge then marked his milestone in style by scoring the winner for the Wanderers, which lifts them off the bottom of the table.
Adelaide needed a win to return to the top four and keep alive its hopes of a top two finish.
Western Sydney continued its rotation policy by making six changes to the side that faced Central Coast Mariners last week.
In a largely uneventful opening 20 minutes, the best chance came from an Adelaide corner with Osama Malik's close range effort cleared off the line by Mateo Poljak.
The Reds again were dangerous from a set piece in the 39th minute. A corner found Bruce Djite completely unmarked in the box. His point blank header was brilliantly saved by Ante Covic.
But it was from a well-worked move on the edge of the area that Adelaide finally made the breakthrough just before half-time.
Tarek Elrich cut in from the right and played a one-two before pulling the ball back. It feel invitingly for Cirio who curled a fantastic shot into the top corner to make it 1-0.
The Wanderers were more attack minded in the second period and their persistance paid off on the hour after a superb pass slipped Mark Bridge through on goal and he was brought down by Nigel Boogaard, who was the last defender.
Boogaard saw red and Tomi Juric slotted in the penalty to make it 1-1 as Western Sydney faced 10 men for the final half hour.
Western Sydney took full advantage of the extra man 10 minutes later.
After the visitor failed to clear a free kick, Juric helped the ball back into the area and Bridge was perfectly placed to fire into the bottom corner and complete the turnaround while marking his 200th appearance in style.
Adelaide tried desperately to get back into the contest, but the Wanderers held firm to record their first victory in a nearly a month.
The Reds miss out on going straight through to a home semi-final and also on to an AFC Champions League berth.
"Of course it's our objective to finish top two and to have a place in the Champions League, but in the end all 10 teams are fighting for the same," Adelaide coach Gombau said.
"So now we have to have a new objective and a new focus. Our focus had to be the finals."
The Reds dominated possession but were clearly lacking the creative flair injured Argentine Carrusca (quad) provides.
After sitting out three games, Gombau hopes to have Carrusca back for the last clash before the finals against Melbourne City next week.
"He is a player that we miss, of course," he said.
"Today we had the ball a lot around his area and we were missing that winning last pass.
"He's a player that makes a difference.
"Next week we hope he can play for a while, and after for the finals we'll have him for sure."
The Wanderers host scandal-plagued Perth in their final game of the season, where a win will ensure they avoid the wooden spoon.
"No one want to finish bottom and I'm pleased that now it's in our hands to make sure we don't," coach Tony Popovic said.
The victory also gives the Wanderers a boost before their crucial ACL game against Kashima Antlers on Tuesday.
With the Japanese side already in Sydney, Popovic hopes the result will send a message.
"Today we didn't look like a team who were bottom. You don't come back like that against a side like Adelaide if you're a poor side," he said.
"So we need to take that into Tuesday."
