Sam Vokes' penalty and a late own goal from Rob Jones gave the visitors a 2-0 win and took them above QPR on goal difference.
Dyche does not have Rangers' funds at his disposal, but the manager is convinced his side can hold their own at the top end of the table.
He said: "When you're top of the league you naturally come over the radar, but we're happy to be a competitive side.
"The obvious challenge is there are bigger clubs with bigger budgets but they don't have a guaranteed license to have success. We can be successful, especially if we keep delivering.
"The secret is, how many times can we do it over a season? We have to keep working every day. At the minute, we have to be relentless and as a group, we want to be relentless.
"The players, from the first moment we got back to pre-season, have taken a lot of information on, worked hard and been open-minded. All we can ask is that they keep delivering.
"Tonight they started better but the game management from us was good and we got on the front foot before finishing strong."
The timing of Vokes' penalty - right on the stroke of half-time - frustrated Rovers boss Paul Dickov, but overall he had few complaints.
"When you get beaten by the better team you have to hold your hands up and not make excuses and that's what happened," he said.
"You have to give Burnley credit. They're a good team, they're top of the league for a reason. I would rather concede a goal late on when we're pushing to score than in another way.
"Burnley did the basic things in football and that was the reason they won the game."