The Dutchman gave the Bundesliga champions the lead in at the Volkswagen Arena, firing home from close range after the hosts had failed to deal with Douglas Costa's low cross from the left wing.
However, Nicklas Bendtner equalised with just over 60 seconds of normal time remaining, thus forcing a penalty shootout that that Dane then decided 5-4 in Wolfsburg's favour, with Xabi Alonso missing from the spot for Bayern.
Robben was bitterly frustrated by the nature of the defeat and conceded afterwards that Pep Guardiola's men have issues to address after losing the showpiece event for the third consecutive season.
“I’m disappointed," the winger is quoted as saying by his club's official website. "That’s football, it can all change in a minute - but we have to look very critically at ourselves because we should have wrapped it up earlier.
"We had our chances and at 2-0 it would have been all over.”
Meanwhile, Bayern defender Jerome Boateng was more upbeat about Bayern's performance, but did agree that the Bavarians had ultimately been punished for their profligacy.
“We shouldn’t be conceding right at the end," the centre-half added. "But, overall, we can be pleased with our performance.
"We were the better team, we bossed the play, had chances and should have killed the game.
"It’s unfortunate and a shame, but I think we have plenty to build on.”
Bayern begin the defence of their Bundesliga title at home to Hamburg on August 14.
