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"It was hard work but I thought we played some good football at times and put Burnley on the back foot and were inches away on several occasions from scoring more goals.
"That's the kind of performance I want the supporters to see every week here. We're channelling our efforts better going forward now and getting more strikes on goal.
"We'll have bad days but it's how you bounce back from that. At the moment we're working hard, creating chances and the excitement is back which I think has been missing."
Burnley made ten changes from the side that had beaten Everton, with players getting their first action of the season. "It was always going to be difficult because most of those lads have been inactive in terms of match time," Coyle said after seeing the Clarets go through.
"That will serve them well for the rest of the season. We trusted the individuals and within the group I thought Diego Penny was outstanding on the night. Fernando can only get better and is an exciting talent, so there's plenty to be pleased about.
"These lads only know how to keep going and give everything they have got and I can't ask any more than that.
"Sometimes you have to earn the right to play and Hartlepool were a credit to themselves. They carved out a very good goalscoring opportunity, they looked bright all night and gave them something to carry on.
"We had a word at half time and freshened things up after ten changes, but in the end I think we got the response we were looking for, showed our quality and in the end they were terrific finishes.
"As befits Burnley there is always a bit of drama and we had a player sent off, but this group show resolve, character and quality and I am delighted to be in the next round."
Coyle said of goalscorer Fletcher: "The two finishes were top class. Steven is a quality player and he's had chances in all the Premier League games. He will continue to get chances and we know what we are getting with him, so he was another big positive tonight."
Burnley ended the game with ten men after Richard Eckersley received two yellow cards. "Richard picked up a yellow in the first half that I thought was soft," Coyle said. "I know why he got a second because he had been pulling the lad and they played advantage.
"I'm not wanting to get into the rights and wrongs of it though. We had to deal with it and we did."