United's Under 18s sealed a well-earned FA Youth Cup win against Middlesbrough thanks to Zak Brunt's sole goal towards the end of the first half.
The Blades came under plenty of pressure from the hosts in the second period of the match, culminating in a 90th minute penalty which Stephen Walker chipped against the crossbar to spare Derek Geary's side extra-time.
It was a tight and tense finale to top off what had been a fairly even match for the most part, the first half in particular had few chances as both teams sought to work each other out.
United had the better of possession and played some good football but it was Middlesbrough who had the half chances, Hayden Hackney firing at Marcus Dewhurst, who was in inspired form, from distance.
Four minutes before half-time and United had the lead, a goal that would turn out to be the decider for the Blades.
A cross from the left found Brunt at the back post and after his initial effort was blocked, the ball did eventually find itself in the back of the net and Brunt was the one wheeling away to celebrate.
The hosts expectedly came out firing in the second half as they looked to save their Youth Cup chances with Derby County waiting in the fourth round.
Walker curled just wide after cutting inside his man before Dewhurst pulled off a superb save to maintain United's lead.
Hayden Hackney fired over when through on goal under pressure from the excellent Kyron Gordon before substitute Stephen Wearne saw his header held by Dewhurst.
The main drama was saved for the final minutes as the game approached stoppage time. With Middlesbrough pressing, substitute Tom Weaver tripped his man inside the box and allowed Walker the chance to send the game into extra-time.
But the Middlesbrough striker chipped his effort against the bar and the referee whistled on the rebound as the ball hadn't touched another player before Walker had another bite at the cherry.
The miss allowed United to hold on and ensure a hard-fought progression to the fourth round and a trip to Derby County.
Blades: Dewhurst, Williams, Boyes, Neal, Gordon, Belehouan, J. Kelly (Foulstone 82), Broadbent, Gaxha, Brunt (Weaver 85), Ompreon (Potts 71).
Unused: Amissah, Gomis.