Tunisian Sami Allagui had given the visitors a 70th-minute lead before Podolski, already a major transfer target for several top European teams for the end of the season, fired in from 22 metres to keep Cologne in 10th place on 21 points.
Mainz, unbeaten in their last five league games, are in 12th place on 18 points.
The match was postponed in November after referee Babak Rafati attempted to commit suicide on the morning of the game.
Mainz were the livelier side throughout but it was the hosts who came closer to a goal just after the restart with Mato Jajalo sent through and keeper Christian Wetklo standing firm to save the Croat's shot from point-blank range.
Tunisia international Allagui, surrounded by three Cologne defenders, headed in a Mario Gavranovic cross past keeper Michael Rensing.
Wetklo was again on hand to palm a thundering Podolski drive on to the crossbar minutes later.
He could do nothing, however, in the 85th minute when the Germany international unleashed an unstoppable left-footer that went in off the post.
The goal lifted Podolski to second place in the Bundesliga scorers' list with one fewer than Bayern Munich's Mario Gomez.
Bayern lead the standings on 34 points from 16 games following their 2-1 win at VfB Stuttgart on Sunday with champions Borussia Dortmund in second place, three points behind after their 1-1 draw at home to Kaiserslautern.
