United's Under 18s fought hard to overcome a Cardiff City side who could have gone top of the Southern Division with a win at the Steelphalt Academy on Wednesday morning.
Jake Young's header inside 20 minutes put the Blades in front before a penalty for the visitors ensured things were level before the break.
It was Tom Wilson's towering header 10 minutes from time that clinched the points on a day where United had to hang on at times.
Derek Geary's side started the better with Tom Williams and Tom Weaver causing issues down the right side, the former putting in a teasing cross which Young fired just wide.
Weaver also put a dangerous low ball in which was cleared away from Josh Scott and it was third time lucky after 17 minutes when Young headed home a Williams cross from close range.
But the hosts struggled to build on their momentum and it was Cardiff who had the better of the game leading up to the break, but a combination of defence and goalkeeper Sam Kelly kept them at bay.
That was until the half hour mark when Cardiff scored, only for the referee to pull play back for a foul and awarded the Bluebirds a penalty which Griffiths duly stepped up to take and slot home.
Cardiff goalkeeper George Ratcliffe pulled off a superb save to claw away a Scott header before getting up to hold a strong effort from Nickseon Gomis from the resulting corner.
As the game got more stretched after the break, the visitors were so close to taking the lead when Sam Bowen saw his free-kick come back off the post and his follow-up effort seconds later strike the other post and go wide for a goal kick.
Ratcliffe once again was in top form down the other end to deny Scott again, but the next corner saw Wilson leap above everyone to head home Frankie Maguire's corner.
Bowen incredibly hit the woodwork for a third time with a powerful long-range effort just before the game moved into stoppage time but United dug deep to hold on for a big three points.
Blades: S. Kelly, Williamson, Foulstone, Wilson, Weaver, Gomis, Antwin (Ackroyd 66), Broadbent, Young (Gaxha 81), Ompreon (Maguire 58), Scott.
Unused: Amissah, Hackford.