The 33-year-old Robben proved he had the legs to sprint clear of the tiring home defence as he capped a superb run with the winning goal in the top-of-the-table clash.
Bayern boss Carlo Ancelotti called the match an "absolutely insane game".
The World-Cup victor would be a good acquisition for Mauricio Pochettino as he would be solid in a back three defensive line-up that the Spurs manager has been favouring for much of this campaign.
Bayern, who have already wrapped up a fifth consecutive league title, trailed 4-2 against second-place Leipzig in the 83rd minute. We may have failed to take points today, but we've won much, much more: hearts and sympathy.
"Leipzig started strongly, but we were the better team towards the end".
Leipzig have been labelled "Bayern-Jäger Nummer 1" which literally translates to "Bayern hunters number 1" in the country and despite being hugely disliked, they are generally seen as the only other team with potential to topple the bavarian giants.
Leipzig's Timo Werner netted twice and with 21 league goals is the Bundesliga's leading German striker, and a shoe-in when Germany's squad for next month's Confederations Cup is announced on Wednesday.
Diamondbacks catcher Chris Iannetta hit in mouth with pitch from Pirates' reliever
More importantly, the Diamondbacks managed a 16-hit attack and increased their home mark to 15-6, the best in the majors. That was until the Bucs' Gregory Polanco homered off of Greinke to break up the no-hitter leading off the eighth inning.
Third-placed Borussia Dortmund drew 1-1 at Augsburg, with Pierre-Emerick Aubameyang scoring his 29th Bundesliga goal of the season.
Marcel Sabitzer had got RB Leipzig off to a flying start scoring in the second-minute, before Lewandowski levelled from the penalty spot in the 17-th minute.
Werner rattled the base of the post but Ralph Hasenhuttl's side by no means had it all their own way and a handball decision when Lahm's cross struck Bernardo resulted in a penalty that Lewandowski sent beyond Peter Gulacsi.
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. The impressive Werner restored Leipzig's two-goal advantage five minutes later. 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.
Dortmund are level on 61 points with Hoffenheim, who crushed Werder Bremen 5-3, but have a better goal difference going into the last matchday next week.
Mainz are effectively safe from relegation after coming from 2-0 down to beat Eintracht Frankfurt 4-2.