The runaway Bundesliga leaders were in ruthless mood in the first half - punishing some sloppy defending from the hosts with Toni Kroos and Thomas Muller efforts.
The Gunners improved slightly after the break and gave their disillusioned fans something to shout about when Lukas Podolski headed home five minutes from the hour to halve the lead.
But that was as good as it got for Arsene Wenger's struggling side, who were always vulnerable on the break and eventually conceded a late third when a Robben-inspired counter-attack provided Mario Mandzukic with a scrambled close-range finish.
Robben, who only came on as a late substitute, told Sky Sports afterwards: "At the moment we are at a very good level. We have a great squad. We improved again compared to last year and are moving in the right direction.
"Tonight, we met a very big step into the quarter-finals but of course we still have a home game to play and we have to stay focus on our own game. We have to try and improve again.
"It wasn't just that we scored two goals but that we expected a very big start from Arsenal here with the fans behind them. I remember it from my time in England.
"We warned each other and that was the message between the players - we have to be there from the first minute because there will be a storm going here at the Emirates.
"I think we did very well. Of course, if you score two goals then you are in a great position."
Bayern will go into the second leg at the Allianz Arena on March 13 as heavy favourites to make the quarter-finals of the competition.