The Whites travelled to face the Clarets on Sunday afternoon in the Premier League and played out a 1-1 draw in Lancashire.
Marcelo Bielsa's side encountered a physical battle, with seven bookings being handed out by referee Michael Oliver in a high-tempo affair.
The two sides were unable to be separated at the interval but it was Sean Dyche's hosts who broke the deadlock on the stroke of the hour.
Former Leeds man Chris Wood was on hand to turn home a low shot from Matthew Lowton after a scramble from a corner inside the United box.
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The visitors responded, though, as they pushed on to earn a draw late on through their No 9.
Bamford - in similar fashion to Wood - was in the right place at the right time to prod a deflected low shot from substitute Jamie Shackleton into the back of the net.
The England man's effort came five minutes from time and ensured Leeds would travel back over the Pennines with a point ahead of the international break.
"It's always tough coming here, you don't want to be disrespectful, but you know what you're going to get when you play against Burnley, " Bamford reflected post-match.
"They do make it difficult and it's a bit of a fight and it was a fight all game. I think we definitely deserved a point, we created a lot of chances but I think we'll be happy with a point to be fair.
"I'd been shooting all day, as well, I thought 'you know what, keep plugging away, keep plugging away', put myself in the right slots and then just a little toe poke from two yards out - I'll take it.
"Sometimes games go like that as a striker. You've got to be patient and not get frustrated even though it's hard.
"Luckily it paid off in the end today, but sometimes you can go a whole game with no chances and it just sometimes ends up like that.
"I think I'm tempted to say that we deserved more, but both teams had spells so I think maybe a point was justified for both teams."
Asked about the physical nature of the clash, Bamford added: "To be honest I was a little bit annoyed with Tarkowski to be fair, it was just a tussle on the floor but he tried to do some weird jiu jitsu thing on me, I don't know what he was doing.
"It was just one of them where - I quite enjoyed having a little fight with them and after the game we were all fine and shaking hands and stuff so it was alright."