A nightmare day for the Blues saw them have skipper Gary Cahill sent off inside the first 15 minutes, allowing the Clarets to race into a 3-0 lead by half time.
The hosts did manage to pull two goals back in the second half but Cesc Fabregas also received his marching orders as Burnley took all three points from their opening fixture.
Dyche's men took until April 29 to register their first win away from home last season and, speaking after the game, the manager praised his players for a "fantastic" result.
"Today big decisions changed the outcome, quite obviously," he said. "The first half pleased me because although Chelsea went to 10 men it was not easy. We kept going, kept probing and it's 3-0 at the interval. And it was weird at half time because you have to decide how you're going to approach it from there.
"Winning away from home on the first day quietens down some stories - we're written off at the start of every season. It kills a few things off early doors. Chelsea have players of such calibre and to see the game through is so pleasing.
"We scored some very good goals but in the second half we felt stuck. Did we expect this situation? No, if we're honest. And Chelsea have top-class players who got themselves back into it.
"It was a fantastic three points and the referee, I just want to say, was fantastic today. Our mentality I was really pleased with, to see the game through. We had to get on, get the job done and I'm please we've done that. We are not naive - we know this is only the start of the season."
Next up for Burnley is a home encounter with West Bromwich Albion next Saturday.