Dutchman Robben, out since Sept. 3 when he was injured in the Netherlands' 1-0 loss to Iceland in a Euro 2016 qualifier, slotted in from close range after a back heel from Robert Lewandowski in the 35th minute, finally breaking the resistance of the visitors.
"I am very happy because this is also a reward for all the hard work," Robben told reporters. "It is a joy to watch (Douglas) Costa play, and Kingsley Coman is doing a great job at 19 years old. We need that quality on the pitch.
"When you can play from the start and then score the first goal then you can be happy," added Robben.
Chile international Arturo Vidal drilled in from Coman's darting run and cutback as the Bavarians extended their record-breaking start to the season to 10 wins in 10 games.
Bundesliga top scorer Lewandowski then got on to the score sheet himself with a glancing header in the 62nd minute, after Costa whipped in a free kick, taking his league tally to 13 goals in 10 games.
Thomas Mueller added their fourth goal with a well-taken penalty as Bayern moved to 30 points, with second-placed Borussia Dortmund, who are on 20, in action on Sunday.
"My first words today belong to Bayern," their coach Pep Guardiola told reporters. "One thousand victories is outstanding and I am proud to be the coach here."
The win lifted any remaining gloom from their first defeat of the season, a 2-0 loss at Arsenal in the Champions League on Tuesday, and kept them on course for a record fourth consecutive Bundesliga title.
Comeback kings Bayer Leverkusen, who scored twice in the final minutes to rescue a 4-4 draw against AS Roma in the Champions League this week, needed only two minutes to overturn a 1-3 deficit and then scored a winner in the 89th with Admir Mehmedi to beat VfB Stuttgart 4-3.
The win lifted Leverkusen to sixth on 17 points one behind VfL Wolfsburg, who edged past Darmstadt 98 with a deflected Daniel Caligiuri shot in the 78th.
Anthony Ujah struck twice as Werder Bremen crushed hosts Mainz 05 3-1 for their first three points in six matches to move up to 14th on 10.
Hertha Berlin climbed to fifth spot, also on 17, after their hard-earned 1-0 victory over promoted Ingolstadt, who were the better side, with Mitchel Weiser's deflected first half goal.
Eintracht Frankfurt put some distance between themselves and the relegation spots with a 2-1 win over Hanover 96 to climb to 12th on 12 points.
(Reporting by Karolos Grohmann; Editing by Clare Fallon and Pritha Sarkar)
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