The Dutch forward struck his 11th league goal of the season in the 69th minute as the Bavarians, chasing a Bundesliga, German Cup and Champions League treble, took advantage of Dortmund's 4-4 draw at home to VfB Stuttgart on Friday.
A scrappy match ended in fortunate fashion for Bayern, who have 60 points from 28 games, when Almog Cohen struck the post for Nuremberg with a long-range effort.
The top two teams have six matches left with Bayern, who have won seven successive games in all competitions, visiting champions Dortmund on April 11.
"The summary of the weekend is that it's all back in our own hands, in all three competitions," said Bayern's sporting director Christian Nerlinger.
"I have great faith in this team. They're showing they can cope with a match every three days and that they're physically in great shape. I'm looking forward to the run-in."
After a scrappy opening hour, Bayern brought on Franck Ribery and Bastian Schweinsteiger.
The winner came when Dominic Maroh gave the ball away just outside his penalty area, Ribery's shot was saved by goalkeeper Raphael Schaefer but Robben pounced on the rebound to score.
Mario Gomez also headed against the crossbar for Bayern who had a lucky escape when keeper Manuel Neuer misjudged Cohen's dipping 30-metre shot and ended up punching it on to the post.
Cologne dropped into the relegation play-off place after losing 2-1 to fellow strugglers Augsburg, heaping the pressure on Norwegian coach Stale Solbakken after a 6-1 home defeat by Dortmund last week.
TWO PENALTIES
Koo Ja-Cheol fired Augsburg ahead in the 19th minute, Lukas Podolski equalised for Cologne with a penalty in the 42nd only for Angola forward Nando Rafael to put the home team back in front with another spot-kick on the stroke of half-time.
"We were very bad, we made big mistakes," said Solbakken.
To make matters worse for Cologne, two other relegation candidates won.
Hamburg SV ended a losing run with a 1-0 victory at bottom club Kaiserslautern and Freiburg continued to climb out of danger by winning 2-0 at Bayer Leverkusen.
Marcell Jansen's 28th-minute goal was enough for Hamburg, the only ever-present team since the Bundesliga was founded in 1963, to end a run of four successive defeats.
Kaiserslautern extended their winless league run to 18 matches.
Julian Schuster and Daniel Caligiuri scored in each half in Freiburg's victory at Leverkusen, making it four wins and a draw in their last five games.
The Leverkusen crowd reacted angrily to the display.
"That was certainly our worst performance of the season," said Dutt. "I can understand the reaction of the fans.
"There is one person responsible for our performa
