The contest was over before half-time as Eugen Polanski, Mohamed Zidan and Nicolai Muller scored in an 18-minute spell in the first half, condemning Cologne to their fourth defeat in five games during which they have conceded 17 goals.
Adam Szalai added the fourth while Cologne forward Lukas Podolski was hit in the face by a small object thrown from the crowd in the first half. He was not hurt and continued playing.
Hertha Berlin also took a step closer to the second division as they lost 2-1 at home to Freiburg with Roman Hubnik scoring at both ends and Augsburg were beaten 3-1 at home by VfB Stuttgart.
Hertha defender Hubnik turned Garra Dembele's shot into his own net after seven minutes and the Malian also set up Freiburg's second for Sebastian Freis after the hour.
Hubnik then headed one back for Hertha who lost for the fifth time in eight outings since 73-year-old Otto Rehhagel took over as coach.
Angola striker Nando Rafael gave Augsburg an early lead from a penalty but it was then all downhill for the promoted side as Serdar Tasci, Martin Harnik and Vedad Ibisevic replied for fifth-placed Stuttgart.
Borussia Monchengladbach, who are fourth which would earn them a place in the Champions League qualifying round next season, had to settle for a 2-2 draw at Werder Bremen despite playing with a man extra for over an hour.
Markus Rosenberg gave Bremen an 18th-minute lead but they had Sebastian Boenisch sent off for a professional foul nine minutes later.
Mike Hanke scored twice after the break to put Gladbach ahead but Brazilian defender Naldo headed Bremen's equaliser in the 74th minute.
Augsburg are 15th in the 18-team table with 30 points from 30 games, one place and one point above Cologne in the relegation play-off spot. Hertha are 17th with 27 points.
PESZKO APOLOGY
Although Peszko has publicly apologised after Saturday night's incident, in which he was widely reported to have become he become drunk and quarreled with a taxi driver who took him to a police station, he was left out of the Cologne team.
Coach Stale Solbakken also omitted Milivoje Novakovic, one of four players left behind when Cologne went to a two-day training camp last week.
"We played reasonably well until we conceded the first goal," said Cologne goalkeeeper Michael Rensing. "Then we suffered four goals again. That for me is inexplicable."
The three-times German champions were quickly in more trouble as Adam Szalai's theatrical fall earned Mainz a penalty which was converted by Peszko's fellow Pole Eugen Polanski in the 18th minute.
Mainz made the most of their luck with two superbly made goals.
Egypt forward Mohamed Zidan drilled the ball emphatically home in the 32nd minute after Muller teed up his shot, then Muller himself got on the end of Eric Maxim Choupo-Moting's measured pass to
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