The Togo striker, who has found life difficult at Eastlands this season, scored twice in the opening 25 minutes, before City were pegged back after the break.
But Adebayor became the first City player to score a European hat-trick and put the game to bed with 17 minutes to go.
City were on the attack early on and Adebayor opened his account with a sublime turn and finish from Patrick Vieira's pass in the 13th minute.
It was two 12 minutes later when the former Arsenal man powerfully headed home David Silva's cross.
But, Poznan got back in the game shortly after half-time when Joel Tshibamba made City pay for some poor defending to slam home in the 50th minute.
Adebayor completed his hat-trick soon after, wrapping things up in the 73rd minute when he latched on to Silva's exquisite pass and prodded home.