Robben, making his first appearance one day after signing from Real Madrid, came on as a half-time substitute for Hamit Altintop with Bayern leading 1-0.
The 25-year-old opened his account in the 68th minute when he latched onto Frenchman Franck Ribery's pass and fired into the corner of the net after a perfectly-timed run.
Ribery was also involved 12 minutes later when the same pair exchanged passes before Robben sidestepped his marker and slotted past goalkeeper Diego Benaglio.
Bayern dominated from early on, Bastian Schweinsteiger heading against the bar before Mario Gomez, the Bundesliga's most expensive signing, pounced on a rebound to score after Benaglio parried Altintop's shot in the 27th minute.
"I couldn't have imagined a better start," said Robben. "The whole team played well together. We controlled the match.
"Franck and I played well together. He's a good player and we need him."
Bayern, who began the weekend 14th in the Bundesliga after their worst start to the season for 43 years, leapt to seventh with five points from four games. Wolfsburg, one place ahead, have six points after their second straight defeat.
Earlier, Schalke 04 suffered a shock 1-0 home defeat against bottom-of-the-table Freiburg while leaders Bayer Leverkusen came from behind to beat Bochum 2-1.
South Korean Cha Du-ri scored Freiburg's winner five minutes before halftime after pouncing on a mistake by Benedikt Hoewedes to hand Schalke their first league defeat under new coach Felix Magath.
Freiburg survived constant Schalke pressure in the second half to hang on for their first win of the season.
Leverkusen fell behind in the 33rd minute when Argentine Diego Klimowicz scored but equalised before halftime when defender Manuel Friedrich headed in following a free kick.
Stefan Kiessling scored the winner from close range in the 68th minute, his fourth goal in as many games, to leave Leverkusen with 10 points from four games.
Hamburg SV, three points behind in second place, host Cologne on Sunday.
Brazilian Carlos Eduardo scored five minutes before halftime to give Hoffenheim a 1-0 win at Hanover 96 while Eintracht Frankfurt were held 1-1 at home by Borussia Dortmund.
