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:
. MIDDLESBROUGH, ENGLAND - AUGUST 14: Middlesbrough player Chuba Akpom celebrates after scoring the second Boro goal during the Sky Bet Championship between Middlesbrough and Sheffield United at Riverside Stadium on August 14, 2022 in Middlesbrough, England. (Photo by Stu Forster/Getty Images)
Chuba Akpom celebrates after scoring the second Middlesbrough goal against Sheffield United. (Photo by Stu Forster/Getty Images) Photo: Stu Forster
. 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
. 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
. 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
. 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
13. Alex Mowatt - 7
(for Forss, 61) Came on for his debut and looked comfortable on the ball while playing some neat forward passes. 7 Photo: Clive Rose
14. Duncan Watmore - 6
(for Dijksteel, 76) A late substitute. Almost got on the scoresheet when he hit the post from inside the area. 6 Photo: Getty Images
15. Matthew Hoppe - N/A
(for Akpom, 90) Made his debut in stoppage-time.N/A Photo: Chuck Burton