Robben scored in the 69th minute, firing to Raphael Schaefer's left after the goalkeeper saved Franck Ribery's initial effort.
Bayern is three points behind Dortmund with six rounds to play. Bayern's goal difference is seven better than Dortmund's, putting the Bavarians' league aspirations in their own hands ahead of the eagerly anticipated clash of the sides in Dortmund on April 11.
"We're not looking at Dortmund, we only want to win our own games," Bayern midfielder Bastian Schweinsteiger said. "You can see the absolute determination of this team ... of course we want the title."
Bottom side Kaiserslautern looks doomed, 10 points from safety, after a 1-0 defeat at home to Hamburger SV, Cologne slipped into the relegation zone with a 2-1 defeat at Augsburg, Freiburg won 2-0 at Bayer Leverkusen, and Mainz won 3-0 at Werder Bremen.
Hertha Berlin was scheduled to host Wolfsburg later Saturday.
Bayern will meet Dortmund in the German Cup final and has one foot in the Champions League semifinal after recording a 2-0 win in Marseille in the first leg of their quarterfinal on Wednesday.
"We have it in our own hands again — indeed in all competitions," Bayern sporting director Christian Nerlinger said.
But his side's hectic schedule appears to be taking its toll as Bayern struggled to create chances against an obstinate Nuremberg side desperate for points after three losses in a row.
Robben had the best chance of the first half in the 34th minute, but the Dutchman was unable to direct his header on target.
It was the closest either side had come to scoring, with neither recording a single effort on target in the first half.
Bayern coach Jupp Heynckes sent on Bastian Schweinsteiger and Franck Ribery early in the second half, aware of the importance of capitalizing on Dortmund's 4-4 draw with Stuttgart on Friday.
Mario Gomez was unlucky to have a goal ruled offside in the 66th, but the visitors found the breakthrough three minutes later when Toni Kroos capitalized on a Dominic Maroh mistake to send Ribery through. The Frenchman fired straight at the goalkeeper but Robben gave him no chance from the rebound.
"We'll probably play badly one day and win," said Nuremberg coach Dieter Hecking, whose side hit the post with two minutes remaining.
Marcell Jansen's 29th-minute effort was enough for Hamburg to ruin Krasimir Balakov's home debut as Kaiserslautern coach.
Kaiserslautern's chances of survival look slim after a club record 18 games without a win.
"We won't give up," Balakov said.
Freiburg inflicted Leverkusen with its fourth defeat in a row, increasing the pressure on beleaguered coach Robin Dutt through goals from Julian Schuster and a fantastic strike from Marco Caligiuri.
In Bremen, Adam Szalai scored off the underside of the crossbar in the 19th, before Eric Maxim Choupo-Moting made it 2-0 after the break and then sealed the win in the 74th.
Augsburg was the only team to win at home, Axel Bellinghausen laying the ball back for Koo Ja-Cheol to deservedly score in the 19th.
Lukas Podolski equalized in the 42nd, converting a penalty he earned under Daniel Baier's clumsy challenge.
But Henrique Sereno brought Baier down at the other end a minute later and Nando Rafael converted the penalty for what proved to be the winner.
"It was a bit like Wild West football," Cologne coach Stale Solbakken said. "I'm disappointed. I take responsibility for this performance which wasn't acceptable for me. But I think I'm the right man (for the job)."