The Scottish Premiership leader made seven changes from the weekend triumph over Ross County after ensuring its progression to the last 32 with last week's 3-2 beating of Standard Liege, but it never looked in danger of failing to secure a fourth win from their six group games.
The new-look side went ahead shortly after the half-hour mark thanks to Itten's first European goal for the club.
It wrapped up just a third win in its past 20 away games in Europe courtesy of Hagi's close-range header with 18 minutes remaining.
Rangers started slowly and were fortunate not to fall behind after 15 minutes, unmarked Poznan captain Lubomir Satka inexplicably heading over from six yards.
The visitor belatedly woke from its slumber and went ahead with its first shot on target after 31 minutes.
Jakub Kaminski sloppily gifted possession to Itten, who waltzed past Djordje Crnomarkovic inside the penalty area before lashing a fierce strike into the roof of Filip Bednarek's net from 10 yards.
Rangers went agonisingly close to doubling their advantage before the interval, but Hagi's deflected strike flashed narrowly past the post.
Hagi had a low shot from outside the area saved by Bednarek early in the second half, while Jon McLaughlin did well to paw away Tymoteusz Puchacz's free-kick at the other end.
Rangers then sealed victory inside the final 20 minutes, Hagi nodding in from almost on the line after Connor Goldson's looping header bounced down off the crossbar.