Sam Vokes (24, 43) netted twice in the first half, either side of a Stephen Ward (39) goal, with Blues captain Gary Cahill having been dismissed for a rough challenge with just 14 minutes played.
Second half substitute Alvaro Morata pulled a goal back on 69 minutes, before another setback for the hosts as Cesc Fabregas picked up a second yellow with nine minutes of the match remaining.
David Luiz was, however, still able to score to set up a frantic final few minutes, but a determined Clarets side held on for a well-deserved three points against the champions.
Dyche told the club’s official website: “It was a fine day overall, for many different reasons.
“I thought we were strong early in the game, while it was 11 v 11. The sending off made a difference and I thought we did really, really well to handle that and keep the ball and create chances.
“We scored three good goals and I think when you are talking about who you are playing against and where you are playing, that’s up there as one of the best 45 minutes of my time here.”