Bayern found themselves 2-0 down to a mediocre Cologne side at half-time in the Munich Allianz Arena and although Belgian defender Daniel van Buyten pulled a goal back towards the end of the match, it was not enough.
Bayern have now lost to Hamburg, Hertha Berlin and Cologne in their last four league games and could drop to fifth in the table on Sunday if Bayer Leverkusen beat Hamburg.
“I believe that after today we must apologise to all our fans for this defeat,” said Rensing, who said last week he was “fed up to the back teeth” with his defenders.
“The first half time was inexplicably bad, Cologne could have won by more. Today was one of those days when nothing went right. We didn't deserve a point, never mind a win. If we had not acted so stupidly, we would already be top of the league; that is irritating.”
First-half goals by Cologne's French midfielder Fabrice Ehret and teenage debutant Daniel Brosinski meant the guests were 2-0 up at half-time and never looked like losing.
It was hardly the ideal warm-up for Bayern's Champions League Round of 16, first-leg tie on Wednesday against Sporting Lisbon in Portugal and chairman Karl-Heinz Rummenigge said it was a bad day at the office.
“Until today, we had played well even in our defeats at Hamburg and Hertha, but today for the first time we played badly,” he said.
“We need to get out of this rut very quickly, starting with Wednesday's game in Lisbon.”
Germany striker Lukas Podolski, making his first start after three months on the sidelines, lasted just 45 minutes before he was substituted.
Podolski was voted the best young player at the last World Cup and will re-join Cologne in the summer. He said anyone of the Bayern team could have been taken off.
“We didn't play well and Cologne deserved the victory,” he said.
“It wasn't just any old Bundesliga fixture for me. I tried my best, but I didn't play well. I accept being substituted at half-time. Any of us could have been substituted, it was not a good afternoon.”
Hertha Berlin lost the top spot of the table on Saturday when they were beaten 2-1 at Wolfsburg, who remains unbeaten at home this season.
A goal from Brazilian Cicero was cancelled out by the two second-half strikes from Wolfsburg's Edin Dzeko to put Berlin down to second.
“We are, of course, very disappointed, because we could have taken at least a point,” said Hertha coach Lucien Favre.
“In the second-half, we were actually the better team and we made Wolfsburg look strong, so it's hard to accept this defeat.”
Hoffenheim went back to the top after a battle royal at Stuttgart as Senegal striker Demba Ba hit a hat-trick which was cancelled out by strikes from Cacau and a pair by Germany forward Mario Gomez, bringing the final score to 3-3.
“The spectators certainly got their money's worth today,” said Hoffenheim coach Ralf Rangnick.
“It was an intense game which would we could have won, albeit unexpectedly. Mario Gomez was impressive today, so we can live with 3-3.”