The Whites toppled their North London visitors 3-1 in the Premier League on Saturday afternoon at Elland Road.
Stuart Dallas opened the scoring from close range to give United a deserved lead before Son Heung-min levelled against the run of play.
Harry Kane saw a goal ruled out a short while later due to an offside flag to turn the game around with VAR showing the decision was a matter of millimetres on replay.
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Bamford, though, put Marcelo Bielsa s men back in front before half-time as he struck home a well worked move from a low Gjanni Alioski cross.
Second half substitute Rodrigo then secured all the spoils six minutes from time with the game stretched and Ryan Mason's side searching for a leveller.
Victory saw Leeds leapfrog Arsenal and Aston Villa into the top half of the Premier League standings to sit in ninth position.
Bielsa's outfit have been in impressive form this term during their debut campaign back in the top flight and with just three games left have more than met their objective of avoiding relegation.
Bamford - who moved his tally onto 15 goals for the year against Spurs - said finishing the season with a bang was something that United's boss wanted from his players despite having very little to play for.
"That was the manager's last speech before we left the hotel," Bamford said of trying to finish the season strongly.
"It was exactly that. We've done really well this season as a team but there's no point going out in a whimper.
"Everyone in football has a short memory, so they just remember the last four games - [the message] was to try and finish the season on a high.
"The first half was stronger for us than the second. They came back into it it a little bit but we stood strong. They are a top team with top players. It's one of those where we had to be fully concentrated.
"That goal at the end settled some nerves I think."
Asked about finding the back of the net for the first time in six games, Bamford added: "In the last few games for some reason we have shored up at the back and not shipped as many chances but we've not created many chances in the last four or five.
"It is not good for me and it's been frustrating but today we got the balance right. Our wingers were brilliant. They put the ball in the right places. They were difficult balls to defend. The [final] goal was coming eventually."