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."
After he had earlier been denied by the post, McCormack headed Leeds in front after Matt Smith had flicked on Connor Wickham's throw-in.
But when Pearce put Kieran Trippier's cross past his own goalkeeper things went from bad to worse for Leeds after the break as Arfield drilled home a winner.
Leeds pushed for an equaliser but one wasn't forthcoming and they slipped to a sixth game without a win, much to the disappointment of manager Brian McDermott.
"It was an improvement but we are desperately disappointed not to have won the game never mind drawn it," McDermott said.
"First half I felt we were the better side and we played how you have to play in this league.
"Unfortunately we gave a couple of goals away which is disappointing but having said that we were much better than we have been in the last two games.
"We got behind them and put them under a lot of pressure. We got to the second balls but from my point of view we didn't get a result and that is the be all or end all.
"Sean Dyche has done an unbelievably good job here as he did at Watford where he was unlucky to lose his job.
"I am pleased for him and you can see the camaraderie that they have here. You have to have that camaraderie and togetherness to be top of the league and that is what we are trying to get."