The first-year defenceman was joined on the score sheet by fellow rookie Mark Scheifele.
After nearly scoring himself, Jets winger Blake Wheeler found Trouba alone in front of the net for the winning goal.
The Oilers forced extra time when Perron redirected a Jeff Petry point shot with two minutes remaining to make it 2-2 on just the fifth Edmonton shot of the third period.
Trouba had tied the game at 5:48 of the third with a short-handed goal when Bryan Little carried the puck down the left wing, went around a sprawling Petry, and found the rookie defenceman in the low slot. It was the league-worst ninth short-handed goal allowed by the Oilers (15-30-6) this season.
Then Scheifele fought through a check along the boards, cut to the front of the net and banged home his own rebound to make it 2-1, his ninth goal of the season, at 10:37 of the third period.
Eberle opened the scoring for the Oilers on the power play at 6:59 of the second period with his 16th goal of the season. Eberle took a pass from Ryan Nugent-Hopkins below the left faceoff dot and beat Pavelec high to the glove side.
The Jets came close in the first 40 minutes, outshooting the Oilers 24-16 through two periods, but couldn't beat Bryzgalov.
The Jets had an 11-8 edge in shots after a scoreless first period that saw each team fail to score on a pair of power plays.
Newly acquired Oilers forward Matt Hendricks took a puck to the ear midway through the period and went to the locker-room, but returned to the game shortly after. Jets winger Blake Wheeler was shaken up after taking an elbow to the head from Taylor Hall, but didn't miss a shift.
Some were expecting the bad blood to carry over from the teams' previous meeting, a 6-2 Edmonton win on Dec. 23 in which the Jets racked up 78 penalty minutes. But there was little action after the whistle until a short third-period scrap between Hendricks and Jets defenceman Mark Stuart.
The Oilers return home for three games, beginning Tuesday against Vancouver. The Jets, meanwhile, face a difficult two-game Californian road trip with games Tuesday in Anaheim and Thursday in San Jose.
Notes: Oilers forward Nail Yakupov, who left Thursday's game with a head injury, was a scratch. Teammate Ales Hemsky missed a second straight game with a foot injury. … Jets winger Evander Kane missed his fourth consecutive game with a hand injury suffered during a fight in the final seconds of a game against Tampa Bay on Jan. 7. … Winnipeg's experiment with Dustin Byfuglien at forward continued for a fourth game. Regularly a defenceman, the 28-year-old Byfuglien was playing in his 500th career NHL game.