United appeared to be heading for victory when Hangeland scored an unfortunate own goal with seven minutes go - the first act of a breathless climax.
Nani missed a penalty moments later that would have put Sir Alex Ferguson's side 3-1 up after Damien Duff handled in the area.
And United were punished for Nani's wastefulness in the final twist as Hangeland atoned for his earlier blunder by powering a header past Edwin van der Sar.
A fine start from the Premier League runners-up was rewarded in the 11th minute when the influential Paul Scholes powered home a low 25-yard drive to register his 150th goal for the club.
But Fulham, who dominated lengthy spells after the interval, showed resolve to claw back the deficit with an emphatic strike from midfielder Simon Davies.
It was a thrilling way for Mark Hughes to mark his first home match as Fulham manager while Ferguson will know his side deserved no more than a point.
Word filtered through shortly before kick-off that Ferguson would continue refusing to conduct interviews with the BBC.
Ferguson was today expected to end his boycott of speaking to the corporation, which dates back to 2004, amid pressure from the Premier League and League Managers Association.
But the United manager is to maintain his stance with assistant Michael Phelan available to provide analysis and comment should the BBC request it.
Ferguson would have been delighted by United's formidable start with a dashing early run from Antonio Valencia caught the eye only for Scholes, the intended recipient of his back heel, to stumble on the ball.
Dimitar Berbatov and Darren Fletcher combined to good effect without any end product as play began to concentrate on Fulham's area.
Only lightening reflexes from David Stockdale kept a header from Berbatov out but moments later the Cottagers keeper was beaten by a magnificent strike from Scholes.
The ball was sprayed around with ease from a corner until Berbatov teed up Scholes who found the bottom right corner with a powerful low drive from 25 yards out.
Davies flashed an effort over the bar at the other end but United were still looking formidable as Berbatov tested Stockdale from a tight angle.
Bobby Zamora was gifted a free header from 10 yards out, but he could only nod a cross by Paul Konchesky wide with the target at his mercy.
Berbatov directed a header over the crossbar but United's defence was in disarray in the 31st minute and only a superb double save from Van der Sar denied Fulham.
Zamora used his strength to cause problems on the edge of the area and possession squirted out to Dempsey, who laid on a perfect pass for Dickson Etuhu.
Etuhu directed his first shot straight at Van der Sar but was presented with a second attempt when the ball rebounded to his feet only for the Dutch keeper, who was sat down, to prod the effort over the bar with an outstretched hand.
Despite the fluidity of United's play, with Berbatov, Javier Hernandez and Valencia prominent, Fulham looked very capable of pinching an equaliser.
Scholes almost jinked his way into a shooting position but Fulham were enjoying the lion's share of possession early in the second half.
Zamora blazed a free-kick over the crossbar after he had been hacked down by Jonny Evans and the balance of power was clearly shifting with the equaliser arriving in the 57th minute.
A rampaging run from Duff swept the winger away from Patrice Era before he found Zamora, whose astute pass fell to an unmarked Davies.
Stood 10 yards out and with time to pick his spot, the Wales midfielder thumped the ball past Van der Sar.
Substitute Nani was making a telling impact and he expertly teed up Berbatov who was bundled off the ball as he was pulling the trigger.
Van der Sar caught a dangerous curler from Danny Murphy while at the other end a Nani free kick crept narrowly over the crossbar before skipper Nemanja Vidic directed a header off-target.
With 11 minutes to go Zamora blew a fine chance, and Fulham were left ruing the opportunity as they conceded a disastrous own goal.
Nani's innocuous-looking corner fell straight to the hapless Hangeland, deflecting off his shin and into the goal.
Fulham suffered another blow in the 86th minute when Duff handled the ball, but Stockdale pulled off a fine save to deny Nani from the spot.
Yet Hangeland and Duff still had time to make amends for their errors in a breathless finish that saw Fulham get what they deserved.
Duff floated in a corner from the right and Hangeland leapt high to nod it home, via a touch from the hand of the diving van der Sar.