The Gunners were saved by a quickfire double salvo from Andrey Arshavin and Park Chu-Young as they continued their unbeaten League Cup run at the Emirates Stadium.
Wenger's side had received a scare at the start of the second half when Bolton took the lead through former Arsenal midfielder Fabrice Muamba.
However, the north Londoners responded rapidly through goals from Arshavin and Park to the delight of boss Wenger.
"The first half I believe we had the chances [to score] but in the second half we were caught straight away in losing the ball in our own half," said the Frenchman, whose side face Chelsea this weekend.
"They scored and from then on we were tested as the more experienced players took charge of the game, like Park, Arshavin and [Yossi] Benayoun. They made the difference.
"I have to congratulate the team as we had two young full-backs who are usually central players that played out of position.
"In midfield we were very young and we showed good spirit against a good Bolton side.
"When we were 2-1 up in the last 20 minutes, our legs had gone a little bit and we had to hang on, relying on a bit of luck because any players had not played at that level at that intensity.
"What was vital for us was not to go into extra-time."
Tuesday evening's match also heralded the return of Thomas Vermaelen after a two-month injury layoff.
The Belgium international has been out with an Achilles injury but started at the Emirates Stadium, playing 84 minutes before coming off with a "tight calf".
"He did very well but he had to come off for a calf problem," Wenger said. "I hope it is not too bad.
"He will be a bit short for Saturday. I do not know yet what has happened.
"We will have to test him tomorrow but it did not look too bad.
"He has been out for a long time and only had one training session yesterday [Tuesday] with the team.
"I didn't expect a game of that intensity because it was a really tough game for a centre-back tonight as they put us under pressure in the second half."
He added: "I don't think it is a calf strain, I think it is just a tight calf because he has not played 90 minutes for three months.
"He wanted to come off. In fairness, I did not want to take him off because I thought it was stupid now with four minutes to go in case we concede a goal.
"He was adamant he could not go on so I hope it is not too bad."
While Arsenal secured progression, they had to withstand a barrage of Bolton pressure to hold out for victory.
Trotters manager Owen Coyle felt disappointed with the result after the match but was proud of his side's effort in north London.
"Arsenal scored two goals from their two shots on target," he said. "They never had another shot on target.
"We scored a very good goal to put us in front and looked to be the team in the ascendancy.
"Then, from my end, the disappointing thing is as good as Arsenal's finishes were, they could have been avoided.
"I thought Arshavin should have been closed down and we paid a heavy cost for that.
"Then we gave the ball away cheaply and got punished by Park with a sublime finish.
"From that moment on it was waves of attacks and not just that but clear goalscoring opportunities.
"We should have buried them, we never did and now we're out of the cup because of it."