Arsene Wenger's surprise decision to start Yaya Sanogo ahead of Olivier Giroud appeared inspired as the pace of the French youngster and recalled Arsenal winger Alex Oxlade-Chamberlain unsettled Europe's tightest defence in the opening stages, only for Mesut Ozil to spurn a ninth-minute spot-kick.
Wojciech Szczesny then brought down Arjen Robben eight minutes before the break at the other end and was sent off, although David Alaba's resulting penalty struck the post for the second missed spot-kick of the game.
Bayern had stunned Manchester City with early goals in both their group-stage games against Arsenal’s Premier League counterparts and Kroos almost repeated the trick when he sent a 25-yard effort arrowing for the top corner in the second minute.
Manuel Neuer stopped Sanogo from giving Arsenal reward for their early endeavour as he produced an excellent reaction stop following Jack Wilshere's prodded pass towards the striker.
Wilshere then released Ozil and Jerome Boateng - who endured a particularly torrid start - felled his Germany international team-mate in the area. But Ozil failed to capitalise from 12 yards as Neuer pushed away a tame effort.
That disappointment checked Arsenal's momentum, although Neuer had to rush out and deny Oxlade-Chamberlain in the 24th minute when the winger got the better of Alaba on the end of Sanogo's raking pass.
Laurent Koscielny then stabbed goalwards from Ozil's 51st-minute free-kick but it was a moment of brief respite and the visitors led three minutes later, Kroos curling into the top corner from 20 yards.
