In doing so, Bamford became the first Whites player in 52 years to score in the club's first three games in a new top-flight campaign with Bamford emulating the feat of Mick Jones in 1968.
But the Whites owed a huge debt to 20-year-old goalkeeper Illan Meslier who produced a string of fine saves including one outstanding effort to deny John Lundstram in the first half though the home side's Aaron Ramsdale also produced heroics in goal.
Playing with a back three of Luke Ayling, Liam Cooper and Robin Koch, Leeds began well with a stinging drive from Ayling from just outside the box beaten away by Ramsdale.
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The Blades then fashioned a chance in the sixth minute as Oli Burke played in Lundstram who dragged his shot wide.
Kalvin Phillips picked up an early booking for a foul in the build up.
After a quiet 15 minutes, a flowing Blades attack midway through the half forced Jack Harrison to make a crucial clearance before a drive from Lundstram was blocked.
United then squandered a decent opening in the 26th minute when a fine free-kick from Kalvin Phillips was headed over the bar by Bamford.
Ramsdale then safely held a deflected effort from Mateusz Klich before a Jack Robinson header sailed well wide for the Blades.
But only an outstanding save from Meslier denied the Blades a 29th-minute opener with the French goalkeeper somehow getting down to keep out a ferocious drive from Lundstram at point-blank range.
Leeds then went close two minutes later when Bamford sent a header over the bar from a fine Harrison cross.
Back back came the Blades with Cooper producing a fine block to a Burke strike before David McGoldrick blazed over the bar.
But a terrific save from Ramsdale then kept out Leeds as Ramsdale managed to parry a scooped effort from Stuart Dallas over the bar and Cooper headed wide from the corner.
In what was becoming a half of fine goalkeeping saves, Meslier then produced another to divert George Baldock's powerful effort wide.
A driving Burke then saw a shot well blocked by Robin Koch and it proved the last real chance of a goalless but entertaining first half.
Record signing Rodrigo was then brought on for Tyler Roberts at the break and Leeds went close to going in front 80 seconds after the restart when a Helder Costa volley was beaten away by Ramsdale.
Nine minutes later, Bamford burst into the box but saw his shot from a tight angle saved by Ramsdale.
Fine defending from Ayling then saw off a dangerous Blades attack before Meslier proved equal to a firm low effort from Enda Stevens.
But Leeds then went their closest yet on the hour mark when Dallas rounded Ramsdale only to see Chris Basham clear his effort off the line and Ayling then fired wide.
After a fine run and cross from Dallas, Rodrigo flashed a header just wide before Ian Poveda was then handed a Premier League debut in replacing Costa.
Cooper then blocked an effort from Baldock before a stinging effort from Rodrigo poleaxed Ethan Ampadu and another curling effort from Rodrigo was then easily held by Ramsdale.
From a rapid Whites counter, Poveda's shot was then saved by Ramsdale but back came the Blades with Baldock cutting inside and seeing his effort deflected behind for a corner which Leeds narrowly survived as Robinson rolled the ball wide.
Brilliant and brave goalkeeping then thwarted Baldock before Ramsdale made yet another save to keep out Dallas.
The derby looked destined to end goalless but in the 88th minute a fantastic cross from Harrison picked out Bamford who headed the ball back across goal to finally beat Ramsdale and record Leeds' second victory from their first three games of the new campaign.
Sheffield United: Ramsdale; Basham, Ampadu, Robinson; Baldock, Lundstram (Norwood 64), Berge, Osborn, Stevens; McGoldrick (McBurnie 74), Burke (Sharp 74). Subs: Foderingham, Jagielka, Fleck, Lowe.
Leeds United: Meslier, Ayling, Koch, Cooper, Dallas, Phillips, Klich, Costa (Poveda 66), Roberts (Rodrigo 46, Alioski 90), Harrison, Bamford. Subs not used: Caprile, Struijk, Davis, Shackleton, Poveda.
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Thank you Laura Collins