The Gunners fell behind to a Robin van Persie header in the first half and although they battled back in the second, Wenger's team failed to force an equaliser.
"We had to give a lot this week - Liverpool, [Borussia] Dortmund and here today. I think a draw would have been a fair result today," he told BBC Sport.
"We conceded a goal on a set piece and in a big game that makes a difference. We rushed our game too much and lost too many balls. In the second half we were composed, had a real go and played our usual game.
"We were very, very, very close a few times. The heart was in the game, the desire was in there, but maybe a bit too much in the first half. We are quite strong on defending set pieces and I was surprised we were caught on the near post."