EB Streymur's extraordinarily busy keeper Torgard smothered a shot from Vassell after the striker connected with a cross from Corluka and saved again from the same player in the fourth minute.
Torgard denied Vassell once more and saved twice from the lively Daniel Sturridge while Elano shot over - all inside the opening 10 minutes.
Streymur played with almost everybody behind the ball and had a real desperation to their defending that gradually frustrated City as the half wore on.
Sturridge created space with an adroit turn but again could find no way past Torgard, who also denied Elano and Gelson Fernandes, while the woodwork came to his rescue when he was beaten by Corluka's header and Petrov's volley.
Streymur were also aided by a timely deflection from a Michael Johnson strike while Micah Richards headed over from a set piece and Bardur Olsen cleared off the line.
City manager Mark Hughes looked less than thrilled as he watched from the Oakwell touchline, while Anghel enjoyed the away team's only decent chance of the half, cutting in from the byeline before shooting narrowly wide from a tight angle.
The home team got the goal they were looking for soon after the restart when Petrov added to his strike in the first leg.
And Elano should have scored after Torgard diverted a cross in his direction but the outstanding keeper made yet another excellent save.
Torgard rushed from his goal to try to clear a long punt forward but was beaten to the ball by Petrov, whose shot missed by inches, while the keeper made yet another save after a strike from Elano took a deflection.
It looked as though City would have to settle for a 1-0 win before Vassell latched on to substitute Ched Evans's flick and skipped round the keeper before slotting home.
Manchester City boss Mark Hughes:
"They defended extremely well. It was damage limitation from them really, but you have to give them praise for their efforts.
"Their goalkeeper had a wonderful night, they threw bodies in the way of everything and made life very difficult for us.
"I was pleased with my lads. They kept at them and if their goalkeeper had not had such a great night, we would have won it at a canter."
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