Pep Guardiola's side rode their luck in a dramatic last 16 first leg clash at the Emirates Stadium, but their class eventually told as Kroos produced his moment of magic and Thomas Mueller bagged the killer second goal in the closing stages.
FC Bayern Munich survived an early penalty miss from Arsenal's Mesut Ozil and a squandered spot-kick of their own by David Alaba.
But Arsenal goalkeeper Wojciech Szczesny had been sent off in the incident that led to FC Bayern Munich's penalty and the visitors eventually capitalised on their numerical superiority through Germany midfielders Kroos and Mueller.
FC Bayern Munich will be expected to finish the job in the second leg on March 11, but Arsenal F.C. can take heart from their 2-0 win in Munich at the same stage last season, even if that result wasn't enough to stop Arsene Wenger's team bowing out on away goals.
"Everything went against us. We should have been 3-0 up before they got a sniff in the game," Wenger said.
"We had more problems in the second half with 10 men because they have very good players. What is for sure is we will have a good go in the second leg."
Bayern, treble winners last season, swaggered into north London widely acclaimed as firm favourites to win both the tie and eventually the competition itself to cap another majestic campaign.
But Arsenal F.C. tore into Bayern with a relish that seemed to shock the Germans and quickly carved out two decent chances, with first Yaya Sanogo and then Santi Cazorla denied by Manuel Neuer.
Arsenal F.C.'s frenzied start had clearly rattled the visitors and they should have been ahead in the ninth minute.