Bullard's ball out of defence found American Dempsey, whose shot was blocked by Anton Ferdinand.
After the initial incident involving Anton Ferdinand, the FA interceded and stripped Terry of the England captaincy.
Anton Ferdinand's injury-time own-goal gave United a point, but it was Wiley's display that incurred Ferguson's wrath.
The visitors took the lead when Cesc Fabregas's block on Anton Ferdinand's clearance flew in from 40 yards.
West Ham brought in Brian Deane, Kevin Horlock, Wayne Quinn and Anton Ferdinand.
Defenders Anton Ferdinand and Phil Bardsley might also have left the field had their three substitutions not already been used.
Anton Ferdinand had an effort ruled out, substitute Peter Crouch headed Spurs back in contention but Boudewijn Zenden's stunning volley sealed it.
Mensah will now compete for a place in the Black Cats' rearguard with Anton Ferdinand, Danny Collins, Nyron Nosworthy and Paulo Da Silva.
Former West Ham defender Anton Ferdinand was dispossessed by Craig Bellamy, who ran through on goal only to be forced wide by Fulop.
Goals from Anton Ferdinand and a Marlon Harewood penalty ensured Alan Pardew's side claimed sixth spot, setting up a play-off semi-final against Ipswich.
Andersen played a superb disguised pass to Berbatov and this time Anton Ferdinand was unlucky as the shot was deflected in off his body.
London's Metropolitan Police is investigating allegations that Terry racially abused Queens Park Rangers defender Anton Ferdinand during an English Premier League match last month.
CNN: FIFA chief Blatter: There is no on-field racism in football
Instead, Albanian midfielder Lorik Cana, deputising at the back with the likes of Nyron Nosworthy and Anton Ferdinand injured, produced a truly disastrous display.
The Blades broke the deadlock six minutes before half-time, Tonge powering an unstoppable free-kick high past Robert Green after Anton Ferdinand had fouled Christian Nade.
BBC: SPORT | Football | Premiership | Sheff Utd 3-0 West Ham
Chelsea and England captain John Terry was charged by UK police after allegedly making racist remarks to Queens Park Rangers defender Anton Ferdinand in October 2011.
CNN: Blatter to enlist Boateng in FIFA's fight against racism
He had opened the scoring at the Stadium of Light with a bizarre goal, when Anton Ferdinand's attempted clearance bounced off the Spaniard and looped into the net.
Former England captain Terry was cleared by a London magistrate's court of racially abusing Queens Park Rangers defender Anton Ferdinand but was then found guilty by an English FA tribunal.
The Suarez affair comes at the same time England captain John Terry is facing a criminal charge of using racist language against QPR defender Anton Ferdinand during Chelsea's EPL game on October 23.
Carew scored his first goal since joining from Aston Villa last month but looked offside as he poked in with only Anton Ferdinand standing between the Norwegian striker and the goal-line.
Most of Birmingham's threat came on the left through Jiri Jarosik, but he could not find a way past Anton Ferdinand and Danny Gabbidon and was restricted to unconvincing long-range efforts.
Sunderland have also been linked with moves for West Ham's Anton Ferdinand and Blackburn's Benni McCarthy and Keane could still add to his squad before the transfer window closes on 1 September.
To make matters worse, Hayden Mullins did not reappear for the second 45 because of a knee injury and Anton Ferdinand failed to see out the match because of a hamstring problem.
Heskey then drew an early yellow card for Kieran Richardson, who brought down the heavyweight striker as he had raced away towards goal after pouncing on Anton Ferdinand's misjudgement of the bouncing ball.
England qualified for the tournament comfortably enough, but Capello quit when the FA removed John Terry as captain in light of a criminal charge of racist abuse against Anton Ferdinand hanging over him.
For the past year, the Blues have been at the heart of a racism row after an incident involving Terry and Queens Park Rangers defender Anton Ferdinand during a Premier League match in October 2011.
The charge is the latest twist in an ongoing saga that began last October, when England and Chelsea defender John Terry was accused of racially abusing Anton Ferdinand, Rio's brother, during a Premier League game.
Coming into the match, Sunderland boss Roy Keane made four changes to the side which lost to Manchester City in their last match with former West Ham pair Anton Ferdinand and George McCartney handed debuts in defence.
Club captain John Terry is currently serving a four-match domestic suspension after the FA found him guilty of racially abusing Queens Park Rangers defender Anton Ferdinand despite the former England international having been earlier cleared in court.
Edin Dzeko leveled two minutes before halftime with his 10th league goal this season, with the Bosnia striker cutting inside defender Anton Ferdinand -- who is at the center of the racism allegations aimed at Chelsea's England captain John Terry -- and slotted in a low shot.
When Hodgson named his 23-man Euro 2012 squad last month he said he omitted the 33-year-old for "footballing reasons, " having been asked if it was because of John Terry's impending court case for allegedly abusing Ferdinand's younger brother Anton.
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