Bamford converted his 10th league goal of the season from the spot in the fifth minute and that was enough to secure his side their sixth Premier League win of the season.
But Leeds, who spurned a host of first-half chances, refused to buckle under the onslaught and inflicted upon Burnley their first defeat in five matches.
Bamford fired his penalty into the top corner to give Leeds an early lead after he had been brought down by Burnley goalkeeper Nick Pope.
Bielsa's side, showing no ill-effects from last week's humbling defeat at Manchester United, threatened again through Rodrigo, but his shot was comfortably gathered by Pope.
It appeared to be a harsh decision and left Dyche and his backroom staff furious.
Both sides were guilty of glaring misses within the space of a minute just after the half-hour mark.
Raphinha wasted another opening, shooting too close to Pope when free in the area and Rodrigo headed off-target following from Ezgjan Alioski's superb cross.
Only a brilliant save from Pope denied Hernandez after a lightning Leeds counter before Barnes fired straight at Meslier at the other end.
Rodriguez fired over at the far post following Brownhill's assist and Meslier produced a fine save to keep out Barnes' stinging effort as Leeds stood firm amid a flurry of late set-pieces.