Robben set up the first goal for Bastian Schweinsteiger and curled in the second himself as last season's Swiss Super League runners-up did little to suggest they could trouble the Bavarians in next week's return.
Odense produced a shock 1-0 win at home to Spaniards Villarreal when Hans Henrik Andreasen scored following an awful defensive blunder with six minutes left.
Wisla Krakow, who like the Danish side are hoping to play in the Champions League for the first time, beat Cypriots APOEL by the same score.
Brazilian playmaker Sammir scored twice to help Croatian champions Dinamo Zagreb to a 4-1 win at home to former European Cup finalists Malmo and Maccabi Haifa beat Genk 2-1.
Bayern have only missed out on the Champions League once in the last 14 seasons and their worst nightmare would be failing to qualify in a season when the final takes place at the Allianz Arena.
Their first-leg victory would appear to be an ample advantage against Zurich whose Letzigrund stadium, with a small capacity and running track around the pitch, is one of Europe's less intimidating venues.
The Swiss took only one point from three home games when they played in the Champions League two years ago.
Zurich's best chance in Munich came after five minutes when Ricardo Rodriguez's long cross found Amine Chermiti and his header flew over.
After that, it was all Bayern and they went ahead three minutes later when Robben broke to the byline and found an unmarked Schweinsteiger who headed in from close-range.
BAYERN CHANCES
Bayern created plenty of chances with Zurich keeper Johnny Leoni producing a superb double save to deny Franck Ribery and Philipp Lahm early in the second half before the Frenchman hit the post with a free-kick.
Mario Gomez, Bundesliga top scorer with 28 goals last season, contrived to fire wide from six metres before Robben curled the second over Leoni in the 72nd minute.
"It was a good result for us and puts us in a good position for the return match," said Bayern goalkeeper Manuel Neuer, who did not have a save to make.
"I'm always happy when the defence does its job."
Villarreal had the better of the match against Odense but the hosts were gifted the winner when Andreasen pounced on a loose back-pass by Colombian debutant Cristian Zapata to fire a rasping drive into the net.
Dinamo, Croatian champions for the last six seasons, dominated their tie and Sammir, who joined the club five years ago as a teenager, fired them ahead in the fourth minute against Malmo with a clinical finish from 10 metres.
Dinamo missed a hatful of chances in the first half, either side of Agon Mehmeti's equaliser against the run of play.
Ante Rukavina restored Dinamo's lead early in the second half, S
