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.
Halftime 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 — which sits at the bottom of the Premier League table and had lost its last 11 trips to Old Trafford — claimed a shock lead in the 19th minute through Steve Sidwell.
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United struggled to break down a resolute Fulham side for long spells, and it was staring down the barrel of a ninth league defeat of the season at one stage as goalkeeper Maarten Stekelenburg excelled.
But Van Persie then swept in Juan Mata's attempted shot from 3 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 of embarrassment, with the manager 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 is now nine points adrift of fourth-place Liverpool, with the result another blow to its 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 ax 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 it sat deep and invited its opponent to attack, but it 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 equalizer, but it was nearly two goals behind as a quick counterattack, led by Tankovic, saw Richardson blaze over when one-on-one with 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 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 away from close range.
The crowd grew more and more frustrated as the home side's consistent decision to cross rarely troubled Fulham.
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.