The Clarets looked set for victory after taking the lead just after the hour mark when former Whites striker Chris Wood flicked home a Matthew Lowton shot as Leeds failed to clear a corner.
But United drew level in the 86th minute when after brilliant work from Raphinha, a fierce shot from Jamie Shackleton was diverted home by Bamford, fresh from his first call-up into the England squad.
Leeds are now on two points for the new campaign following last weekend's 2-2 draw at home to Everton and a home clash against Liverpool is next after the September international break.
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Leeds and 'keeper Illan Meslier initially stood up to a series of high balls from Burnley towards the Whites box including a corner from which Diego Llorente tangled with Wood who ended up on the deck.
Burnley had the ball in the net as Ben Mee walloped home a loose ball from a free-kick but the whistle had already gone for a foul on Luke Ayling.
Moments later, Meslier raced off his line to save well from Wood and Johann Gudmundsson squandered a good chance when curling a shot wide shortly after the half hour mark
Leeds, though, squandered a glorious opening in the 35th minute when a flowing move started by Stuart Dallas ended with Raphinha firing wide from a tight angle.
Another rapid counter a few moments later was also wasted as Kalvin Phillips played in Bamford whose touch was too heavy and the chance was gone.
Back came Burnley and Ashley Barnes fired in a vicious effort on the turn that Meslier turned behind for a corner from which Wood headed over.
From yet another Leeds break, Bamford's cutback was cleared before a firm effort from Harrison was blocked from a corner.
But the Clarets were only denied by the frame of the goal on the stroke of half time as a corner deflected on to the post before Leeds cleared.
Burnley's Dwight McNeil had the first attempt of the second half but sent his shot from range well wide four minutes in.
Ten minutes later, Josh Brownhill went much closer as an effort from similar range whistled just past the post and another attempt from McNeil flashed wide just after.
From a corner, a James Tarkowski header then smashed against the crossbar but Leeds failed to clear their lines and the Clarets went ahead in the 61st minute when a Lowton shot was flicked on by Wood and Meslier spilled the ball over the line.
Leeds tried to respond but neither Bamford or Raphinha could get a shot away from a scramble in the Clarets box.
Shackleton and Tyler Roberts were sent on from the bench as Diego Llorente and Rodrigo made way but Leeds struggled to create chances and looked destined for a defeat.
But with four minutes left, Raphinha cut inside from the right flank and sent in a powerful attempt that was saved but only cleared to Shackleton whose rocket of a shot was diverted home by Bamford at the back post.
Leeds then went all out for a winner and forced late corners but the final opportunity of the game was headed wide by Phillips to conclude a full blooded contest at Turf Moor.
Burnley: Pope; Lowton, Tarkowski, Mee, Taylor; Gudmundsson (Lennon 63), Westwood, Brownhill, McNeil; Wood, Barnes (Vydra 79). Subs not used: Hennessey, Bardsley, Pieters, Collins, Thomas, Cork, Rodriguez.
Leeds United: Meslier, Llorente (Shackleton 65), Cooper, Struijk, Ayling, Dallas, Phillips, Rodrigo (Roberts 72), Raphinha, Harrison, Bamford. Subs not used: Klaesson, Cresswell, McCarron, Forshaw, Summerville, Costa, Gelhardt.
Referee: Michael Oliver.
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Thank you Laura Collins