Europa League - Adebayor treble fires City to win
The striker pounced to give City the lead on 12 minutes as he accepted Patrick Vieira's pass on the edge of the box before turning to slip the ball into the corner of a net with a delicious finish.
Joel Tshibamba gave the visitors fresh hope when he thumped home a volley after some sloppy defending by City five minutes into the second period.
The play raged from end to end, but City completed a merited victory when Adebayor converted the outstanding Silva's pass on 73 minutes from five yards out.
City move three points clear of Poznan on seven points at the top of Group A after three games.
Italian giants Juventus were held to a 1-1 draw by Salzburg in the section's other match of the night.
City travel to Poland for the return leg on November 4.
Mancini is keen to see his side reach the final of the competition, but the Premier League takes priority over everything else at the moment.
Poznan gave a good account of themselves, but were ultimately undone by a lack of real quality in key areas.
Slawomir Peszko fired wide of goal on three minutes before Adebayor's double seemed to settle the home side as the teams headed up the tunnel for the half-time break.
A fair number of the several thousand Poznan fans in the crowd were bouncing around without any tops on, but they were warmed up when Tshibamba struck after Joleon Lescott appeared to fell Peszko in the area.
Poznan suddenly sensed the chance of an unlikely point, but they blew a golden chance when Tshibamba hammered over the bar with just Joe Hart to beat from 10 yards on 68 minutes.
Adebayor made the visitors pay with a clinical finish three minutes later to complete the hat-trick.
Silva departed to a deserved standing ovation from the home fans on 72 minutes when he was replaced by Yaya Toure.
City could have added to their tally when Adam Johnson showed off his considerable skills to dance beyond Luis Henriquez before rattling in a shot that came back off a post, but the three points were already in the bag.