The Germany international was reprimanded by his compatriot and Gunners captain Per Mertesacker for not applauding the travelling supporters at the end of the Premier League game and heading straight to the dressing room.
"Sorry I didn't thank the fans at the end of the game!" he wrote on his official Facebook page. "You have been brilliant to me and I know you had travelled a long way and spent your money to support us.
"I was upset with the result and know I should have come to you to say 'thank you' and I know it is a big Arsenal tradition win, lose or draw."
Arsenal are currently two points clear at the top of the league table.