The home side dominated the first half and Dallas fired them into a 13th-minute lead when he swept a close-range rebound into the roof of the net after Spurs goalkeeper Hugo Lloris parried a mis-hit clearance.
Heung-min Son levelled in the 25th minutes with a classy finish after Dele Alli put him through with a defence-splitting pass but Leeds were unfazed and Bamford restored their lead shortly before halftime.
It was a sweeping move down the left as Jack Harrison released Ezgjan Alioski and Bamford tapped in the defender’s cutback.
Tottenham striker Harry Kane had two goals disallowed for offside and the visitors missed several chances before late substitute Rodrigo sealed victory for Leeds when he capped a fast break with a clinical first-time shot in the 84th minute.
Additional reporting by Reuters.