The Clarets hit the front just after the hour mark when Matt Lowton's shot found the net via the boot of Chris Wood, but having dropped deep and allowed Leeds to regain control, Burnley conceded on 86 minutes.
Jamie Shackleton linked up with Raphinha, who tricked his way past Charlie Taylor brilliantly and saw his cross cut out by Ben Mee, Shackleton picking up the ball and sending it goalwards, a deflection off Mee, then Patrick Bamford's outstretched foot, beating Nick Pope, not in dissimilar style to the opener.
Dyche was pleased with how his side managed against Leeds and their style of football.
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"Tricky side to play against Leeds, they want it open, they want it like a basketball match," he said.
"I think we dealt with that quite well, they had a couple of breakaways which you expect.
"We started well in the second half and took the game on. I don't like to excuse goals but I have to say, Chaz [Charlie Taylor] gets an unlucky trip, Ben gets a great block and then it goes in. I think it was very unlucky."
Burnley picked up their first point with the 1-1 draw, Leeds moving onto two points after last week's 2-2 draw at home to Everton.
"Concern doesn't change things, performances change things," he said.
"We have to find the balance, work hard to make it come in our favour. There wasn't a tremendous amount wrong with our performance. We still need to find that cutting edge a little bit better in their box. It's a very unfortunate one. Someone just sent me a thing saying 'Leeds add to Burnley's woes', but they're only a point ahead of us, it's a funny thing, this early in the season.
"I don't think there's been much of a wrong direction. We were decent against Brighton, Liverpool is a tough ask but I thought the performance was decent. Today as well I thought the performance was decent. Leeds are a tricky side to play against, it's not just us, everyone finds them tricky to play against but I thought we managed it well and our shape was good. We lost our way a little bit for that period when they scored, but without them really opening us up. They get a bit of a stuffy goal, don't get me wrong we'd all take one and hopefully we will too over the course of the season."
Referee Michael Oliver showed his yellow card seven times during the game, which had niggle to it right from the off when Bamford tangled with James Tarkowski.
Marcelo Bielsa saw nothing more than a 'disputed game' and Dyche echoed his sentiment.
"It was just a competitive game," said the Burnley boss.
"I think the game had gone too far the other way, I think everyone recognised that. In the tournament over the summer the fans enjoyed a more balanced view of the game and not every time a player's touched someone going down on the floor screaming and yelping, I think they enjoyed that that's gone away.
"I think it's positive they said we should let the game flow a bit more. We all want to see technical quality, tactical understanding and the physical prowess of some of these players but I don't think there's anything wrong with competing as well, it's part of any sport and certainly football.
"It was two teams working very hard to win a game."