Dutchman Robben, who scored the winner against Dortmund in the Champions League final last season, tapped in at the far post in the 107th minute after the country's top two teams slugged it out in a goalless yet entertaining 90 minutes.
Mueller then raced clear deep in stoppage time to complete the win over their bitter rivals immediately after Marco Reus had missed a chance to equalise.
Victory ended a bad run for Bayern who had won the league title with seven games to spare in Pep Guardiola's first season in charge. They were eliminated in the Champions League semi-final by Real Madrid and lost 3-0 at home in the Bundesliga to Dortmund.
Saturday's final was marked by a controversial goalline incident with Mats Hummels' header in the 64th minute cleared on the line by Bayern defender Dante with television replays showing the entire ball may have well crossed the line.
German football bosses earlier in the year voted against the introduction of goalline technology but the latest incident is set to renew the debate.