The Blades were handed a second successive defeat as Brazil forward Richarlison struck early in the second half at Bramall Lane.
Chris Wilder's side remain in eighth place, two points behind seventh-placed Wolves who host Crystal Palace later on Monday.
United's hopes of capping a memorable first season back in the Premier League with a Europa League berth will be over if Wolves beat the Eagles.
The teams who finish fifth and sixth are certain to reach next season's Europa League.
If the FA Cup winners finish in the top six, then seventh place would also qualify for the Europa League.
Arsenal, who face third-placed Chelsea in the FA Cup final, are currently 10th.
Everton striker Dominic Calvert-Lewin went closest to breaking the deadlock in the first half as his header struck a post.
Carlo Ancelotti's team won it in the 46th minute when Richarlison met Gylfi Sigurdsson's free-kick with a glancing header for his 15th goal of the season.