The hosts looked nothing like a team battling relegation for much of the season, scoring at will as Mainz suffered their third loss in five games since the winter break.
Cologne moved two points clear of the relegation playoff spot into 13th place on 25 points with Mainz stuck in fifth on 37.
In the only other game on Sunday, Werder Bremen salvaged a 1-1 draw against fourth-placed Hanover 96 to stay just above the danger zone on 24 points in 14th place.
Didier Ya Konan gave the visitors the lead against the run of play with his 11th goal of the season midway through the first half before Bremen, the better side throughout, deservedly levelled with a Per Mertesacker header in what was Bremen coach Thomas Schaaf's 400th game in charge.
Borussia Dortmund lead the standings on 52, 10 ahead of second-placed Bayer Leverkusen.
Cologne's Podolski gave the hosts a fortuitous lead on three minutes when his 35-metre free-kick, intended for team mate Martin Lanig, sailed undisturbed into the net.
Mainz levelled on the half hour through Sami Allagui but Slovenian striker Novakovic scored two minutes from the break to put Cologne back in the driving seat.
Podolski and Novakovic struck again in a five-minute spell early in the second half before Petar Sliskovic netted a late consolation.
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