Welbeck was taken off at halftime in his comeback game.
“It was planned before the game that he would only play 45 minutes," said Wenger.
“He has only just come back from injury, and we have three more games on Sunday, Wednesday and Saturday again.
“He has no problem. Of course, he was ready to stay on and frustrated to come off, but he's in good shape."
On the defeat, Wenger admits their play had been sluggish.
Wenger said: "We missed the accuracy of our passing in the final third. Cologne defended with fantastic spirit and determination.
“It took us some time to inject pace into the game. I cannot fault the concentration and preparation of the players."
He added: "We had a lot of possession. We lost the game to one shot, basically. It was a very nice penalty that we have to take on board. It was honestly not a penalty at all.
"We missed a little bit of the accuracy in our passing in the final third that is required to score more goals. We had plenty of dangerous situations but we always missed a little something to finish our chances. I must also say that Cologne defended with spirit, with commitment and overall the crowd was absolutely fantastic for them."