Heskey opened the scoring in the second half and, after Clarke Carlisle had equalised for the visitors, Downing's stunning strike settled Wednesday night's fourth-round tie at Villa Park six minutes into extra-time.
Villa boss Houllier, who twice won the League Cup with Liverpool, said: "I don't take credit for the subs. I wanted to rest them.
"But, with other players tiring, they made a difference.
"Subs have to be in the right frame of mind, have the right attitude.
"If the hunger is there, they can make a difference. Their freshness can bring initiative - and they did that.
"Emile came on and changed the game."
Houllier, who lost John Carew through illness before the game, was full of praise for Burnley.
He said: "Credit to them, they played well. We made it difficult. We should have scored at least one in the first half.
"That affected our confidence. We didn't play as well as we should have done in the second half.
"They really wanted it, credit to them. We really wanted it too, which is good."
Villa winger Marc Albrighton will miss the derby with Birmingham on Sunday through suspension after a professional foul on Wade Elliott.
Houllier said: "Mark's sending off was a fair decision. He didn't touch him (Elliott) a lot but I've no complaints."
Burnley boss Brian Laws felt it was an injustice that his side has gone out of the competition.
Laws said: "I've got to say, football is cruel at times, and tonight for us it was. We didn't deserve the result.
"The performance was outstanding, with terrific play from us at times. We are very unfortunate not to be in the hat for the next round.
"We were beaten by an outstanding strike (by Downing) and it needed to be as Lee Grant was terrific tonight.
"The hero of the night was probably Brad Friedel. He certainly earned his corn tonight with some great saves at the end.
"All I've got is praise for my players and the performance is a real sign of what we can do."