Robben rocket rescues Bayern
The Dutchman found the net from 25-metres with a wind-assisted shot in the 65th minute to make the aggregate score 4-4 after the four-times European champions won the first leg 2-1.
Fiorentina's Juan Manuel Vargas took advantage of a Hans-Joerg Butt fumble for the opener in the 28th minute and Stevan Jovetic made it 2-0 for the hosts in the 54th before the home crowd went on an emotional rollercoaster ride that ended in disappointment.
Mark van Bommel levelled the tie for Bayern in the 60th before Jovetic bagged his second, only for Robben to decide it with his splendid strike almost immediately after.
"It was a tough match, especially because of the conditions," Bayern midfielder Sebastian Schweinsteiger told Sky Italia.
"We were clever to keep the ball on the ground. At the end of two legs we have shown ourselves to be stronger."
Fiorentina will rue their back luck from the first leg when a late Miroslav Klose goal scored from an offside position condemned them to defeat.
Ending Bayern's 18-match unbeaten run will be meagre consolation, especially as they are unlikely to qualify for the competition next season after a miserable run of form that has dragged them down into the middle of the Serie A standings.
The hosts, without much missed Romania striker Adrian Mutu after he failed drugs tests and Massimo Gobbi for his red card in the first leg, tried to dictate the pace from the start.
The rain and strong wind made keeping the ball under control difficult but the Italian side went ahead when Marco Marchionni's shot was fumbled by Butt and Peruvian winger Vargas pounced to score.
Bayern were stirred into action and Fiorentina goalkeeper Sebastien Frey produced a remarkable save to deny Robben six minutes later.
Alberto Gilardino should have extended Fiorentina's lead in the 49th when he blasted straight at Butt with the goal gaping before making amends by laying on Jovetic's first goal with a back-heel.
Van Bommel rifled in from outside the box to level the tie on the hour.
Jovetic was on target four minutes later after being set up again by Gilardino but it took just 72 seconds for Robben to reply decisively for the Bundesliga leaders.
"There is a lot of anger and disappointment. Tonight there are no regrets as we put in a great performance. It's the result of the first leg which hurts," Frey told Sky Italia.
"To play in these conditions with the wind against was tough. But Robben scored a great goal."