Sunday's Yorkshire derby looked destined to end in a goalless draw but only after heroics from both Blades 'keeper Aaron Ramsdale and 20-year-old French shotstopper Meslier who produced one particularly outstanding save to deny John Lundstram in the first half.
With time running out, Leeds then bagged the only goal of the game in the 88th minute as Bamford connected to a precise cross from Harrison to finally beat Ramsdale by heading the ball back across goal and into the left corner.
Bamford, though, felt he should have netted in the first half when putting another header over and the 27-year-old forward saluted the talents of both Harrison and Meslier when speaking to BT Sport moments after the full-time whistle.
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Informed that Meslier had been named man of the match, Bamford said: "I'm not surprised.
"Every shot that he saved their players were like 'who is this guy?'
"But he is brilliant keeper.
"He is only young. I think the first time you lot saw him was actually against Arsenal and you saw how good he was then and he is just growing with each game.
"He deserves all the plaudits, he works hard and he is keeping a level head so he deserves it."
Bamford has now netted in all three of his side's first three games of the new top-flight campaign with the Whites no 9 becoming the first player in 52 years to achieve that feat with no player having scored in Leeds' first three games of a top-tier campaign since Mick Jones in 1968.
"This one came off my nose as well but I'll take it!" smiled Bamford.
"But to be honest I was a bit disappointed that I didn't score with the header that went over in the first half.
"It's important that we got the three points and obviously I am happy to score.
"It was a tough game.
"I think importantly for us we got a clean sheet after the first two games. But they made us work for it.
"The way they play it's very hard to come up against and to press like we want to press is difficult against their style of play.
"He has got them really well drilled and it was a tough game."
Hailing the crossing of winger Harrison, Bamford said "To be fair, before the game, I said to Jack, 'every time you get the ball I'm going between the middle centre back and the far one so put the ball there.'
"In the first half he did it and I didn't score and then in the second half he did it and I got the goal so fair play to him for putting it on the money."
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Thank you Laura Collins