The Gunners had fallen behind to a Fabrice Muamba goal, but stuck back through Andrey Arshavin and Park Chu-Young to book their place in the next round.
Coquelin said that the team did not play well enough in the first half against Bolton, but were inspired by Arsene Wenger's words during the break.
“First of all I am really pleased to win this game. I thought tonight was good, Bolton put out a strong team. We had a young squad out there today but I think we've done really well coming out with a 2-1 win - it's great,” he told the club’s official website.
“They put a lot of pressure on us and scored the goal so after that we had to play better. We didn't play very well in the first half and we didn't move the ball around quickly so in the second half the manager asked us to do that. That's what we did.
“We got caught but I think we did a good job in coming back into the game. Andrey scored the first one five minutes after and it helped us.
“I think we showed great character. Obviously some players were making their debut like Nico Yennaris, who had a terrific game tonight and played really well. Ignasi Miquel as well, and I think everyone played really good.”
When asked about specific players, Coquelin was just as effusive, adding: “You know Andrey and you know he is a great player. He has got the touch, he can get goals, assists and I think he played really well for us tonight. He helped us a lot. You could see that his touch was amazing.”
And on youngster Yennaris, he remakred: “I think he is more of a midfield player but he can play at right-back. Tonight I think he did a terrific job and hopefully things will carry on going for him that way.”
Coquelin made his fifth appearance of the season against Bolton, with Wenger showing faith in him since the departures of Cesc Fabregas and Samir Nasri.
And the midfielder says that he is thrilled to have been afforded so much time on the field this season as the Gunners look to respond to a slow start to the campaign.
“It's great. I've been playing a few games - big ones as well - and every time the manager needs me I just try to give my best and be the best I can be on the pitch,” he said.
And on playing alongside Emmanuel Frimpong, he added: “Obviously it is a bit strange. I was talking to Jack [Wilshere] the other day and was saying to him that is he wasn't injured it could have been the same three in midfield as it was two years ago [in the 2009 FA Youth Cup run].
“But it was great to play with Emmanuel, I have known him since I came to Arsenal so it makes things easier."
Arsenal are now in the last eight of the League Cup once again, having made it to Wembley last year, only to lose in the final to Birmingham City.
But Coquelin believes the character shown by the side against Bolton means that they are capable of going one better this time around.
When asked if the Gunners could win the competition, he replied: “Yes. We showed character tonight and with that we can go pretty far in this competition and win it.”
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