Marcelo Bielsa's free-flowing Whites again created a host of goalscoring chances at Saturday's Championship hosts with 21 shots on goal as opposed to Luton's 11 with 12 on target compared to five from the Hatters.
High-flying Leeds finally made the breakthrough six minutes after the break when Bamford applied a tidy finish to a counter-attack from a Ben White through ball but fourth-bottom Luton levelled just three minutes later through a James Collins header.
Despite creating further chances, United looked destined to head back to Yorkshire with just a point but in the final minute of normal time, a right flank cross from Mateusz Klich was diverted past the excellent Luton 'keeper James Shea by Pearson under pressure from Bamford.
Bamford, though, said there was no doubting he would be claiming the strike as his own with the forward insisting he had applied the initial touch to a goal that sent Leeds back into the automatic promotion places.
After putting himself on seven goals for the season even if the official records will say six, Bamford also played down any concerns about United leaving it nail-bitingly close - and late - to seal an important victory which put them second.
"After the game to be honest you don't really think about it because you've won the game so that's the main thing," said Bamford, asked about the missed chances prior to 'his' winner.
"It's always nice in a game to have more than a one-goal cushion.
"But I suppose in a way it also helps keep us focused knowing that we have still got to attack.
"Whilst you would like to win by a lot more, the three points is what's important."
Reflecting on who had actually scored the winning goal, Bamford smiled: "I stood across the front and I think I actually got my studs on it.
"I didn't have a clue where it went so I didn't have a clue it had gone in until I looked up and I thought there's 'no chance I'm letting them think it was an own goal!' I'm running off claiming it.
"I don't know whether it was heading towards goal but I got a nick on it. As a striker you claim them!"
Bamford could easily have had a hat-trick with the striker unlucky to hit the post in the first half after a superb Gaetano Berardi through ball.
The Whites no 9 also fired a good chance over in the second half while Bamford and United in general were continually denied by Luton 'keeper Shea who pulled off a string of fine saves.
But plucky Luton also had their moments with former Whites loanee Izzy Brown pulling the strings and setting up their equaliser with a fine cross for Collins to convert.
The Hatters also missed a glorious chance in the fourth minute of added time when Dan Potts sent a free header over while the home faithful were incensed with United's opening goal with Brown claiming to have been fouled before United's swift counter attack.
Moments after Collins had equalised, the Hatters also saw another effort netted from a goalmouth scramble disallowed for offside.
Bamford, though, agreed with Whites head coach Marcelo Bielsa in that United were fully deserving of a victory that sent them second and three points clear of third-placed Preston North End after their 1-0 loss at Derby County.
"Before the game we said we need to aim for three points today," said Bamford.
"Without being cocky or disregarding the talent and potential that they have, I think we knew our style of play would favour us against them and I think you saw that.
"We dominated the whole game and pushed them back further and further.
"They were good on the counter attack so we had to keep our wits about us but in terms of chances I think they only had the goal and obviously the one that got disallowed and then the one right at the end but I think a win for us was a fair result. It probably could have been by a greater margin."