Ignacio Scocco and Nikos Lyberopoulos netted in the space of three minutes midway through the first half to set up the Greek side's victory.
Rovers, who also went down 2-1 to Rangers in their opening game on Sunday, pulled a goal back late on through centre-back Phil Jones but it was more than they deserved in a match completely dominated by AEK.
The diabolical conditions meant only a small crowd turned up but they were treated to some good football for most of the 90 minutes, especially from the Greeks.
It was AEK's second win in as many games in the tournament, meaning a draw with Scottish champions Rangers on Sunday will assure them top spot in the table.
Rovers manager Sam Allardyce made five changes to the side that lost to Rangers on Sunday - including starting Australia international Brett Emerton on the right side of midfield - but it made little difference as the Greeks controlled the early exchanges.
AEK were finding plenty of joy down their right and it was no surprise it was an attack down that channel that led to the opening goal on 21 minutes.
Nathan Burns raced onto a long pass from Pantelis Kafes and beat Jones for pace before cutting the ball back for Kafes who produced an audacious flick pass to Lyberopoulos. He controlled and teed up Scocco, who beat one defender and lashed a low shot past Paul Robinson into the far corner.
Scocco turned provider three minutes later, receiving a pass from Burns on the right before his wicked centre found its way to Lyberopoulos who produced a sensational finish on the half-volley past a stranded Robinson.
Both sides lost their way in the second half as both managers made a raft of substitutions but Rovers pulled one back with a rare shot on target eight minutes from time.
The AEK defence went to sleep at a free-kick and allowed substitute Morten Gamst Pedersen to float in a cross that was headed home by Jones.