After pushing Liverpool all the way in a 4-3 defeat on the opening weekend of the season, Marcelo Bielsa's men have won their next two with the much-maligned Bamford scoring in all three games so far.
The Blades remain without a point or a goal this season after impressing to finish ninth on their return to the top-flight last season.
Elsewhere, West Ham United got its first league win of the season in style with a 4-0 win over Wolverhampton Wanderers, despite the absence of manager David Moyes because of coronavirus.
Moyes was still taking charge from home as he was in communication with his staff and will have enjoyed what he saw as Jarrod Bowen curled them into a deserved half-time lead.
Bowen doubled his tally on the rebound after Pablo Fornals hit the post before Raul Jimenez's own goal and Sebastien Haller rounded off an impressive performance from the Hammers.