Lee came off the bench to head the winner a minute from time after goals from Johan Elmander and Kevin Davies had been cancelled out by Cameron Jerome and Kevin Phillips.
It will be Bolton's first semi-final appearance in the competition for 11 years, but Coyle is aiming higher.
Coyle said: "We are into the semi-final of the best domestic competition in world football.
"It is a terrific achievement. (Bolton goalkeeper) Jussi Jaaskelainen said to me, `I've waited 12 years to get to this gaffer, Wembley in the FA Cup'.
"It is nice when you have players of that calibre really cherishing those occasions. But what you want now is to progress to the final. Whatever the draw, we will be ready to go and try and win that game when it comes about."