Brewer, a former Oiler, scored 2:04 into overtime as the St. Louis Blues extended Edmonton's losing streak to a disheartening 10 games with a 3-2 victory.
Petr Cajanek and Jeff Woywitka also scored for the Blues (30-29-12).
Kyle Brodziak and Shawn Horcoff countered for Edmonton (30-35-7), which has been outscored 37-9 during its streak, which is one loss short of the club record. The Oilers also haven't won in nine games since dealing Ryan Smyth to the New York Islanders at the NHL trade deadline.
Injuries are certainly a factor in Edmonton's misfortune, considering the club is missing 11 players. And the news didn't get any better Saturday as the Oilers had just four defenceman for most of the game after an injury to Ladislav Smid.
"It has not been an easy time by any stretch of the imagination," lamented Oilers coach Craig MacTavish. "I don't think any team could overcome the amount of injuries that we have had.
"When you have 11 guys out of your lineup what are you supposed to do? I've been reasonably satisfied with the way we've played positionally. I haven't been unhappy. So I'm not even close to slitting my throat."
And MacTavish said he didn't feel his team deserved the victory.
"We didn't play well," he said. "I was more forgiving of our effort the other night (a 2-1 loss to Minnesota on Thursday) than tonight.
"It was pretty shoddy at times. Dwayne Roloson played great in net and gave us the opportunity to win. But outside of that we didn't play very well."
Blues head coach Andy Murray agreed his team deserved the extra point.
"The Oilers being handicapped with the defence situation played its part but I think we were the deserving winners," he said.
Blues assistant captain Doug Weight, another former Oiler, was glad to see his team finally come away with a win in its third straight overtime game.
"It was good to get four points out of six on this trip," he said. "We played pretty good and had some great chances and (Roloson) played well.
"That team showed a lot of heart and obviously took us to overtime and we were fortunate to get a good bounce in overtime there."
Cajanek opened the scoring with just 2:20 left to go in the first period, blasting a shot past Roloson for his 13th goal of the season.
Edmonton made it 1-1 just 40 seconds later when Brodziak picked up a loose puck in front of the Blues goal and beat goalie Curtis Sanford with a shot through his legs for his first career NHL goal.
Woywitka's first career goal put the Blues back ahead 13 minutes into the second. Oiler rookie defenceman Brian Young's attempt to redirect a pass went right to a pinching Woywitka, who converted the miscue.
Edmonton tied it again halfway through the third as Horcoff scored the Oilers' first power-play goal in 32 attempts.
The Oilers play the third game of their six-game homestand Monday against Vancouver while the Blues are home to Ottawa on Tuesday.
NOTES: As badly banged up as the Oilers are, the Blues had seven players out for Saturday's game a Horcoff returned from a bout with the flu to replace Joffrey Lupul, the latest Oiler felled by the bug . . . Petr Nedved's second stint with the Oilers has ended. Nedved, knocked out of the lineup earlier this month with a knee injury, has left town in a move agreed upon by both himself and the team. The 35-year-old will be an unrestricted free agent this summer . . . St. Louis rookie David Backes has really picked it up offensively since early February. In the 14 games prior to Saturday's contest, he had eight goals and three assists. Prior to his hot streak, Backes had one career NHL goal in 22 games.