The German was one of six players recalled by Gunners boss Arsene Wenger and he struck a quickfire double before the half-hour mark to help them establish a 2-0 lead at the interval.
Arsenal were dominant in the opening stages, although their first goal came on the break as Mesut Ozil slipped the ball through for Podolski to round the keeper and tuck home on 15 minutes.
An individual effort from Carl Baker nearly drew Coventry level as the Sky Blues captain travelled 40 yards before unleashing a thumping long-range shot which Lukas Fabianski tipped over the crossbar.
The League One outfit soon found themselves further behind, though, as Podolski nodded home at the back post after compatriot Per Mertesacker had won the flick-on from a Gunners corner.
A wild shot on his weaker left foot flew high into the stands as Arsenal's goalscorer wasted a good hat-trick chance on 40 minutes. Serge Gnabry also missed an opportunity to increase his side's advantage before half time, but Wenger's men are still firmly on course to reach the fifth round.