Ross McCormack opened the scoring as Leeds threatened to hand the Turf Moor faithful their first taste of defeat but Jason Pearce headed into his own net to level the scores at half-time.
Scott Arfield drilled home on 67 minutes to hand Burnley all three points as they extended their unbeaten league run to 14 games, with Dyche believing they can keep this run going if they let loose.
"It was an interesting game and I warned the players as I knew Leeds would be up for it. I think we were up for it but we just gave a soft goal away," said Burnley boss Dyche.
"I never believe in just defending a game, I believe in attacking. I never had so much quality as a player but I had a relentless mentality and I believed in it.
"I have told the lads they can achieve anything. I have told them they have to be relentless and I thought yet again when things weren't going our way, we just kept going.
"We can play better but we still looked a threat. It is one game at a time as we know the division is relentless so we have to be relentless in our performances to win the league.
"It is hard making your full debut as Ashley [Barnes] was doing, we all know that. In the second half we saw what he was about, he was willing to work and his hold up play was really good. We take great value in him and think he will do very well for us."
- Sports & Recreation