Robben, unhappy about his substitute role this season, only made the starting lineup for the hosts due to the absence of the suspended Franck Ribery but capped a sparkling performance with his 43rd-minute curled shot.
Wednesday's victory was sweet for treble hunters Bayern, looking for their first silverware in three years, as they took revenge for their 5-2 German Cup final loss to Dortmund last season.
"With this win we have reclaimed our top status in German football and the balance has been restored," a beaming club president Uli Hoeness told reporters.
The Bavarians, who had lost five of their last six games against their rivals but hold a 17-point lead over the champions at the top of the Bundesliga, were the better side throughout.
Dortmund, who won the German double last season and the league in 2011, realistically now have only the Champions League title to challenge for.
"We were that bit more aggressive and really wanted to win this game," Bayern coach Jupp Heynckes told reporters. "It was intensive but we were one goal better. We also had a lot more chances in the first half."
Heynckes' team are also eyeing a spot in the Champions League last eight after beating Arsenal 3-1 away in the last 16 first leg earlier in February.
Robben shot out of the blocks and twice came close in the opening stages in front of a 71,000 crowd which included UEFA president Michel Platini and German national coach Joachim Loew.
The hosts turned up the pressure and Bastian Schweinsteiger chipped a superb pass for Javi Martinez whose powerful volley was punched clear by Dortmund goalkeeper Roman Weidenfeller in the 36th minute.
Robben finally managed to beat Weidenfeller, who made a dozen stunning saves to keep Dortmund in the game, with a trademark 20-metre effort from the right.
"We just could not get our game going although we tried everything," said Weidenfeller, who also came to the rescue late in the game, saving two consecutive Dante headers at point blank. "Maybe we were just not fresh enough."
Dortmund, badly missing central defender Mats Hummels, were left exposed at the back far too often while they also failed to find ways to punch through a solid Bayern backline with striker Robert Lewandowski - a reported Bayern transfer target - having a bad day.
The final whistle saw Robben kneeling on the pitch, his fists clenched and celebrating wildly by himself.
"We really wanted to show today how good we currently are," Robben told reporters.
Earlier on Wednesday Bundesliga club VfB Stuttgart edged past gutsy second tier VfL Bochum 2-0 to book their spot in the last four.
Christian Gentner grabbed the lead for Stuttgart, who last won the trophy in 1997, when his shot from about 18 metres took a wicked deflection to sail over keeper Andreas Luthe and into the net in the 18th minute.
Bosnian striker Vedad Ibisevic ended his goal drought stretching back to December by drilling the ball in from 11 metres in the 81st minute. He has now scored in every round of the Cup this season.
On Tuesday, Freiburg came from two goals down to beat Mainz 05 3-2 in extra time to advance, along with VfL Wolfsburg who edged past third tier Kickers Offenbach 2-1.
The semi-final draw takes place on Sunday.
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