Davies made 11 changes at Turf Moor as they faced one of the Championship's form sides in the third round.
Davies was encouraged by the display of some of his fringe players.
He said: "It was a good cup tie and I thought we ran a very good Burnley side very close. I thought, in the first half, that we were the better side.
"But they then got an equaliser through what was a deflection, which fell to the edge of the box. Then it is a wonderful finish. In the second half, it could have gone either way, I am sure you will agree - because we should have been 2-1 up ourselves in the second half.
"When it was 1-1 the next goal, in my opinion, was going to win the match. Unfortunately, a player who has done fantastically well for us, in Matt Derbyshire, did not take his chance (for a second goal).
"It was a good cup tie and we ran them close. I said to the players at the end there that it is little things which lose you games of football. If we had taken the lead again - that was a big moment.
"Certainly there were one or two good performances and we are still a side that is going very well. We gave Burnley a good game."