Robben seals Bayern's 22nd German league title.
Bayern had already opened up an effectively insurmountable three-point lead over title rivals Schalke and ended the season five points clear as the players celebrated in style at Berlin's Olympic Stadium.
Having also scored twice on his debut last August, Robben's double means he finishes with 16 league goals for the season.
The 26-year-old cost Bayern 24 million euros from Real Madrid, but has been worth every penny after a stellar season that saw him play a huge part in getting Bayern to the Champions League final against Inter Milan on May 22.
"It is important that we won here," said Bayern coach Louis van Gaal, who was drenched in beer by his players after the match in what has become a Bundesliga custom.
"We have the best defence, the best attack and deserved to be crowned Bundesliga champions. It is good that we were victorious to keep the winning rhythm. Tonight, we will celebrate in style."
Croatia striker Ivica Olic put Munich ahead when he hit his 11th league goal of the season on 20 minutes by drilling Robben's pass into the bottom-right corner.
Berlin fought back when Adrian Ramos levelled on 59 minutes before Robben curled home a shot on 74 minutes and then added his second with a simple tap-in three minutes from time.
Bayern will be back at the Olympic Stadium next Saturday when they play Werder Bremen in the German Cup final.
Schalke were already assured of second place and an automatic place in next season's Champions League. They finished their campaign with a 0-0 draw against Mainz.
Goalkeeper Manuel Neuer, who is set to be Germany's number one at next month's World Cup, saved a penalty from midfielder Miroslav Karhan on 75 minutes after bringing down Andreas Ivanschitz.
Bremen will have the chance to join Schalke and Bayern in the Champions League after finishing third in the league thanks to a 1-1 draw with Hamburg, which entitles them to a place in the play-off round.
Peru striker Claudio Pizarro opened the scoring on 59 minutes with a superb strike from a pass by Clemens Fritz before Hamburg's Dutch striker Ruud van Nistelrooy equalised in the 82nd minute.
Bayer Leverkusen qualified for a Europa League place for next season after a 1-1 draw at Borussia Moenchgladbach saw them finish fourth.
Borussia Dortmund will join them in the old UEFA Cup competition as they finished fifth despite losing 3-1 at Freiburg.
Stuttgart, who flirted with relegation before the arrival of Christian Gross as manager last December, completed their fairytale comeback when they finished sixth to claim the last European place with a 1-1 draw at Hoffenheim.
Defending champions Wolfsburg completed a disappointing season with a 3-1 win over Frankfurt that left them eighth in the table.
In the fight to avoid the drop, Hanover, whose captain Robert Enke tragically committed suicide last November, preserved their Bundesliga status with a 3-0 win at Bochum that sent their opponents into the second division.
Nuremberg beat Cologne 1-0, but it was not enough to save them from a relegation showdown.
Bochum and bottom side Hertha will now be relegated and Nuremberg finish in 16th, which means they will face second division Augsburg in a two-legged play-off next Thursday and Sunday for the right to a place in the Bundesliga.
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