Two goals in as many minutes from Robin van Persie and Michael Carrick looked to have secured a vital win for the champions, only for the visitors to fight back at the death
Half-time substitute Bent headed in during the fourth minute of added time as United's comeback, led by late goals from Robin van Persie and Michael Carrick, was ultimately denied.
Fulham – who sit bottom of the Premier League table and had lost their last 11 trips to Old Trafford – claimed a shock lead in the 19th minute through Steve Sidwell.
|VIEW FROM OLD TRAFFORD
|By Peter Staunton
Well, Manchester United thought they had got away with it. Despite dominating the territory and the possession stats, they were justifiably pegged back as the league's bottom-dwellers came away with a share of the spoils.
Darren Bent's late goal was as unexpected as it was deserved. Fulham's defensive fragility threatened to undo 75 minutes of hard work as they capitulated, giving United two goals they scarcely deserved.
More questions for David Moyes who will be asked why the champions cannot close out a home match against the worst side in the division.
But Van Persie then swept in Juan Mata's attempted shot from three yards at the far post in the 78th minute and, less than two minutes later, Carrick's deflected strike from outside the penalty area sailed into the top corner.
That seemed to have swung the game and spared United boss David Moyes' blushes, with the Scott celebrating the second United goal emphatically on the pitch.
Fulham, though, produced a late sucker-punch for under-fire Moyes, as Kieran Richardson's shot – which was parried by David de Gea – was nodded in by Bent.
United are now nine points adrift of the fourth-placed Liverpool, with the result another blow to their hopes of qualifying for next season's Champions League.
After announcing he will quit Old Trafford at the end of the season, Nemanja Vidic returned from a three-match suspension to captain the side, while Fulham boss Rene Meulensteen swung the axe with a bold team selection after four consecutive league defeats.
The former Manchester United assistant manager gave new signings John Heitinga and Ryan Tunnicliffe their Fulham debuts and 18-year-old forward Muamer Tankovic made his first Premier League start.
However, there was no place for fellow recent signing Kostas Mitroglou, while Brede Hangeland and Scott Parker were only on the bench.
Fulham's intention was obvious from the initial stages, as they sat deep and invited their opponents to attack, but they took a shock lead on a rare foray forward through Sidwell.
Lewis Holtby, making just his second appearance since arriving on loan from Tottenham, lofted a perfect ball into the penalty area for Sidwell, who did not have to break stride before cushioning a volley into the bottom corner.
Van Persie shot over and Carrick volleyed wide as United pressed for an equaliser, but they were nearly two goals behind as a quick counter-attack, led by Tankovic, saw Richardson blaze over when one-on-one with goalkeeper De Gea.
Fulham's Stekelenburg then made a superb save at the other end, keeping out a fierce Carrick effort that made its way through a crowd of bodies, and Van Persie's follow-up was turned wide for a corner – from which Vidic headed straight at the Netherlands keeper.
Wayne Rooney drew an outstanding save from Stekelenburg nine minutes into the second half, with the Fulham man showing great reactions to turn the ball over from close range.
The home crowd grew more and more frustrated as their side's consistent decision to cross rarely troubled Dan Burn and fellow Fulham centre-back Heitinga.
Van Persie's great positioning finally drew United level though, and Old Trafford then erupted when Adnan Januzaj's cross was not cleared properly and Carrick struck, via a deflection.
However, Bent had the last laugh, reacting quickest after Richardson – who was played in by the excellent Sidwell – saw his shot parried to end a thrilling contest.