Robben, 33, then proved he had the legs to sprint clear of the tiring home defence as he capped a superb run with the victor.
Bayern coach Carlo Ancelotti: "It was an incredible match".
"We had a hard start, but at the end it was enough for this insane result".
"They are playing Champions League football next term and that will be a great experience".
"Congratulations Bayern for this mad victory and congrats to those who were in the stadium to witness it", said Leipzig coach Ralph Hasenhuettl. We may have failed to take points today, but we've won much, much more: hearts and sympathy. Significant changes in the team did not happen, Bayern remained in his already played composition, which is approaching the inevitable championship.
Bayern's Robert Lewandowski, second left, challenges for the ball with Leipzig goalkeeper Peter Gulacsi, third right, during the German first division Bundesliga soccer match between RB Leipzig and FC Bayern Munich in Leipzig, Germany, Saturday, May 13, 2017.
Hosts Leipzig restored the lead when Xabi Alonso fouled RB's playmaker Emil Forsberg and Werner netted home the spot kick.
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Leipzig looked to have the match sewn up as they led 4-2 in the 84th minute, but Robert Lewandowski pulled one back to set up a grandstand finish before Alaba and Robben produced moments of real quality to seal a remarkable turnaround.
But Bayern stayed in touch as Thiago Alcantara headed home Alonso's cross on 60 minutes, before Werner grabbed his second on 65 after turning Germany defender Jerome Boateng.
Thiago Alcantara halved the deficit on the hour mark, but Werner quickly restored the two-goal cushion to nearly certainly give Leipzig the win. Questions will be asked of Bayern's defence, which shipped four goals for the first time since losing 4-1 at Wolfsburg in January 2015.
"No, I have no problems with Franck", he told a media conference previewing Saturday's clash against second-placed RB Leipzig.
Augsburg dented Borussia Dortmund's hopes of claiming the final automatic Champions League place by holding on for a 1-1 draw.
Mainz did their chances no harm by coming from behind to beat Eintracht Frankfurt 4-2.
The results of the last rounds were not without surprises and, although Leipzig remains for 7 games in a row without defeat, until victory in the last round over Gertoy, Leipzig played a draw with Ingolstadt and Schalke.