Marouane Chamakh headed the hosts back in front, with Gary Cahill then sent off for a tackle from behind on the Moroccan.
The unbeaten Gunners have now won three of their opening four league games, making the most of rivals Manchester United dropping points at Everton while also making sure Chelsea, with a 100% record, are still within touching distance.
"We have had two games, at Blackburn and today, which we certainly would not have taken six points from two years ago," Wenger said. "The players can handle these situations.
"Overall it was a competitive game where we dominated and created many chances, but when you play against Bolton, as long as you don't kill the game off, they can score - you can dominate 80% of the game and still lose.
"They marked us very tight, which provoked a very physical game, but they also played well.
"Bolton had full commitment and we needed to be patient, to get our technique and passing to prevail.
"It was a good test for our two centre-backs, who did very well and I am pleased with the performance of the whole team."
Referee Stuart Attwell deemed Cahill's challenge from behind worthy of a straight red, but also let some other hefty challenges go unpunished - most notably Paul Robinson on Arsenal substitute Abou Diaby, who latter hobbled off.
Wenger reflected: "There could have been other red cards.
"You have to see it again because in the heat of the game it is sometimes difficult to judge, but the challenge on Diaby was a bad tackle.
"He is not in a very good shape. Diaby cannot move his leg at the moment, so we will see how we get away with it."