Niko Kovac's side had been locked in a pulsating race for the title with Borussia Dortmund but went into the final game of the season knowing a point would almost certainly see it retain the title.
Kingsley Coman set it on its way after just four minutes, although the visitors levelled entirely against the run of play in the 50th minute through Sebastien Haller.
Parity did not last long, however, as David Alaba and Renato Sanches scored in quick succession before Ribery and Robben came off the bench to wrap up the title in style in the final 20 minutes.
Coman ran on to Thomas Muller's pass and slotted past Kevin Trapp from 12 yards for his sixth Bundesliga goal of the campaign, and Bayern could have been out of sight within the opening 15 minutes had Trapp not produced a hat-trick of excellent saves to deny Serge Gnabry, Alaba and Robert Lewandowski.
Gnabry had a strike ruled out by VAR (video assistant referee) after Lewandowski had strayed into an offside position in the build-up, while Jonathan de Guzman crashed against his own crossbar as Bayern continued to pile forward at every opportunity.
Those missed chances came back to haunt Bayern five minutes after the restart, when half-time substitute Haller prodded home from after David Abraham's effort had bounced back off the crossbar.
Normal service was resumed just three minutes later, however, when Alaba raced in to tap home after Trapp had kept out Muller's effort from distance.
Sanches, who had replaced the injured Leon Goretzka in the first half, then put the result beyond doubt after 58 minutes, cutting in from the left and firing through Trapp for his first Bundesliga goal.
Ribery then signed off from Allianz Arena in trademark fashion in the 72nd minute, jinking between two defenders and clipping superbly over Trapp for an emotional fourth.
Not to be outdone, though, Robben tapped in from Alaba's cross 12 minutes from time to kick-start the celebrations.