He punished the home side in the ninth minute from 12 yards; Darius Vassell's pass across the box finding the midfielder after Michael Johnson had intelligently let the ball run through his legs.
City doubled their lead in the 28th minute when Hamann spectacularly connected with Petrov's corner for his first goal in 95 matches.
Joe Hart, largely unemployed in the City goal during the first half, was called into action after the break as a more adventurous Streymur tried to push forward.
But they could not work their way through a City side which had started with nine internationals.
Hughes will be disappointed City were not more prolific in front of goal, but the return leg at Oakwell on 31 July should be a formality.
The Welshman will also be pleased that defenders Micah Richards and Nedum Onuoha came through the match unscathed.
The duo were making their comebacks from injuries, with Richards absent from the City line-up since February because of a knee injury and Onuoha missing since April with a shoulder problem.
Manchester City manager Mark Hughes:
"I was pleased with the performance, particularly in the first half when we kept possession well.
"It could have been more comprehensive but two goals away from home in Europe and a clean sheet is a good result."
Of summer signing Jo, Hughes added: "He showed great awareness. Once he gets to know the players and the way we want to play he will be a big asset to us."
EB Streymur:
Torgard, Bo, Clementsen, Djurhuus, Foldgast (Davidsen 85), Bardur Olsen, Jacobsen (Brian Olsen 76), Hans Pauli Samuelsen (Eliasen 77), Hanssen, Arnbjorn Hansen, Niclassen.
Subs Not Used: Magnus Poulsen, Balog, Kruse, Thomassen.
Man City:
Hart, Onuoha, Dunne, Richards, Ball, Ireland, Hamann (Gelson 74), Johnson, Petrov, Vassell, Jo (Evans 74).
Subs Not Used: Schmeichel, Corluka, Elano, Mwaruwari, Castillo.
Booked:
Johnson.
Goals:
Petrov 9, Hamann 28.
Att:
3,000.
Ref:
Nicole Petignat (Switzerland).
