It was party time at Bramall Lane as second-half goals from Sander Berge and Anel Ahmedhodzic wrapped up second place for the hosts - and ensured a play-off place is surely beyond Carlos Corberan's ailing Baggies.
A landmark evening brought up 5,000 league fixtures for Albion - just the sixth English club to do so - but there was only one dressing room celebrating.
Corberan's visitors looked good for the best part of an hour and were the better team in the first half, mostly thanks to the returning Taylor Gardner-Hickman, but his haphazard backpass allowed the Blades to open the scoring.
From that moment there ws no way back. They failed with any return from a game in hand and while only two points from sixth with two games left, the volume of rivals means long-fading dreams have all-but disappeared.
Corberan's sole change saw Gardner-Hickman introduced to the starting line-up in place of John Swift, who dropped to the bench.
And it was like-for-like for the Baggies as Gardner-Hickman, goalscorer from the bench at Blackpool last midweek,
The Blades made three changes from the side seen off 3-0 by Manchester City at Wembley in Saturday's FA Cup semi-final via a Riyad Mahrez hat-trick.
Tommy Doyle and James McAtee, City loanees, returned to the Blades midfield while Oliver McBurnie replaced Daniel Jebbison up front, playing up with dangerman Iliman Ndiaye.
Yokuslu was fine to feature again four days after playing 85 minutes against Sunderland on his way back from an ankle injury. Kyle Bartley was again among the substitutes on his way back to fitness.
The Bramall Lane atmosphere was cranked right up ahead of kick-off with an air of expectancy among the packed-out home stands.
But the setting didn't intimidate Corberan's men, who settled well and were the better side in the opening stages.
Albion winger Karlan Grant, without a league goal since late August, showed why that might be the case with a poor miss inside two minutes.
Jed Wallace's cross from the right to the back post was a peach and although the bounce was tricky and a defender added pressure, Grant couldn't make decisive contact to even hit the target.
It served as a warning to the Blades but one they hardly heeded and bright Albion were denied a clear penalty, so showed TV replays, on 15 minutes. Brandon Thomas-Asante was clipped for a free-kick in what was revealed as just inside the right corner of the box.
The free-kick was eventually squared to Grant who tested keeper Wes Foderingham from the edge of the box. Foderingham was up quickly to save a rebound that was flagged offside.
The hostility cranked up inside the stadium as referee Dean Whitehouse and his officials somehow missed a nailed-on home corner before Erik Pieters was booked for a very strong challenge. Sheffield United settled with the ball but struggled to force anything on Alex Palmer's goal.
Pieters' late challenge drew a yell of "that was worse than Santos!" from an irate Blade - referencing Georges Santos' hack down of Andy Johnson - on commentary to his right in the press box - which sparked the battle of Bramall Lane 21 years ago.
Defender Jack Robinson's harmless poke at Palmer was the best the Blades had to offer in the first period.
The Bramall Lane faithful grew frustrated as Albion used the ball well and at times were patient, while other times countered well.
Gardner-Hickman was bright and acrobatically connected with a Wallace cross for a good bicycle kick effort, but Foderingham held on.
Heckingbottom's side started the second period with a bit more gusto but were still limited to strikes from distance early on.
The Baggies looked composed but, in a split second, a momentary lapse of concentration made all the difference.
There was little sense of danger as Gardner-Hickman was given possession 45 yards out from his own goal towards the right. But, without taking a touch, his first-touch backpass was riddled with danger - and so it proved.
In raced Ndiaye. He was tracked but showed expert composure and squared for Berge in the penalty area with Palmer stranded. Furlong make a burst to get back but couldn't quite make up the yards and Berge duly converted for Bramall Lane to erupt.
A firecracker exploded inside the stadium as Albion imploded. The wind was taken from their sails and the Blades grew.
John Egan somehow nodded wide before Corberan introduced Swift and Bartley for a shift in momentum. It didn't work and by 76 minutes the contest was done.
A corner was only half-cleared and George Baldock's return was lifted into the roof of the roof of the net by Ahmedhodzic.
It was game and almost-certainly season over for Albion.
Sheffield United (3-5-2): Foderingham; Ahmedhodzic, Egan (c), Robinson; Baldock (Bogle, 90+1), Doyle (Norwood, 80), Berge (Basham, 90+1), McAtee (Fleck, 74), Lowe; Ndiaye, McBurnie (Sharp, 74).
Subs not used: Davies, Jebbison.
Albion (4-2-3-1): Palmer; Furlong, Ajayi, Pieters (Bartley, 71), Townsend (c); Molumby, Yokuslu (Swift, 71), Gardner-Hickman Wallace (Malcolm, 90+3), Grant (Albrighton, 78); Thomas-Asante.
Subs not used: Griffiths, Rogic, Livermore.
Referee: Dean Whitestone