The Dutchman struck his 11th league goal of the season in the 68th minute as the Bavarians, chasing a Bundesliga, Cup and Champions League treble, took advantage of Dortmund's 4-4 draw at home with VfB Stuttgart the previous evening.
A scrappy match ended with a let off for Bayern, who have 60 points from 28 games, when Almog Cohen struck the post with a long-range effort.
The top two have six matches each to play with Bayern, who made it seven wins in a row in all competitions, visiting defending champions Dortmund on April 11.
Nuremburg and Bayern produced a scrappy opening hour although the game came to life in the final third after Bayern brought on Franck Ribery and Bastian Schweinsteiger.
Their winner came when Nuremburg's Dominic Maroh gave the ball away just outside his own penalty area, Ribery's shot was stopped by goalkeeper Raphael Schaefer but Robben collected the rebound and scored from the edge of the area.
Mario Gomez headed against the crossbar for Bayern, who then had a lucky escape when Neuer misjudged Cohen's dipping 30-metre shot and ended up punching it on to the post.
Cologne dropped into the relegation playoff place after losing 2-1 fellow strugglers Augsburg with all the goals coming in the first half.
Ja-Cheol Koo 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 Augsburg back in front with another spot kick on the stroke of halftime.
To make if worse for Cologne, two other relegation candidates both won.
Hamburg SV ended a losing run with a 1-0 win at hapless bottom club Kaiserslautern and Freiburg continued their impressive climb out of danger by winning 2-0 at Bayer Leverkusen.
Marcell Jansen's 29th-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 Bayer Leverkusen, making it four wins and a draw in their last five games for the visitors.
Freiburg climbed to 13th place with 31 points from 28 games, one ahead of Hamburg and Augsburg.
Eric Maxim Choupo-Moting scored twice in Mainz 05's 3-0 win at Werder Bremen after Adam Szalai had opened the scoring.
