United took the lead inside three minutes when midfielder Sander Berge found space inside the box and beat Boro keeper Zack Steffen with a low shot.
The hosts responded well and drew level in the 14th minute when Chuba Akpom converted Isaiah Jones’ low cross from close range.
A Ryan Giles own goal then put the Blades back ahead before Akpom scored a second with eight minutes remaining.
Here’s how each Boro player fared:
1. Zack Steffen - 6
Was close to stopping the opener when he was beaten by Berge’s low shot.Made a fine save to deny Ryan Brewster in the second half. 6 Photo: Hector Vivas
2. Anfernee Dijksteel - 7
Swept up danger and read the game well. Was a little hesitant in the build-up to United’s second goal. Subbed in the second half as Boro searched for an equaliser. 7 Photo: Getty Images
3. Dael Fry - 6
Made his first Championship appearance of the season in the middle of Boro’s back three. Made some important interceptions as the visitors pressed high up the pitch. Replaced at half-time. 6 Photo: Getty Images
4. Darragh Lenihan - 5
A mixed afternoon. Wasn’t tight enough to his man for both of United’s goals but assisted Akom’s second with a powerful header. Almost scored the winner with a late header. 5 Photo: Alex Livesey