The game was then settled by Victor Moses's right-footed shot into the bottom corner of the net on 69 minutes, which kept Chelsea's top-four hopes alive.
Conte was pleased with how Chelsea played against a difficult opponent, and also praised the two-man strikeforce of Alvaro Morata and Olivier Giroud, which gave Burnley trouble throughout Thursday's game.
"We played a good game against a difficult team because it is not simple to play against Burnley," Conte told BBC Sport.
"We played the game very well and the two strikers worked very well. They played a fantastic game with and without the ball. The whole team played an intensive game. We deserved to win.
"In the past it was very difficult for me to try this option with the two strikers but they showed a great link and a great combination and at the same time they worked very hard for the team.
"For sure we will try to reach the FA Cup final. It won't be easy but we must be ready to fight. I saw a fantastic spirit tonight. We must continue this way."
Chelsea, who trailed fourth-placed Tottenham Hotspur by 10 points last week, have now narrowed the gap to five points.