Arsenal left it late to snatch victory over Fulham after they had fallen behind as a result of goalkeeper Jens Lehmann's horrendous error.
Hleb came up with an unlikely winner to rescue Arsenal
With only 50 seconds gone, Lehmann made an awful mess of clearing Gael Clichy's backpass and Healy tapped in.
Lehmann then had to watch his opposite number Tony Warner produce countless fine saves to keep Arsenal at bay.
But Robin van Persie smashed home a penalty on 84 minutes before Alexander Hleb slotted in a last-gasp winner.
It had looked to be Fulham's afternoon for so long before the late strikes as Warner - a last-minute replacement for Antti Niemi, who picked up an injury in the warm-up - was outstanding.
In the first half, he made fine stops from two low Tomas Rosicky shots, which had the keeper scrambling across goal, and saved his best to deny Hleb and Rosicky when both were clean through.
And as the first period drew to a close, Warner came rushing out of his goal to use his legs expertly to prevent Cesc Fabregas putting the final touch to a superb Hleb through-ball.
Later, Hleb felt he should have had a penalty when Chris Baird appeared to trip him but his appeals were waved away.
Fulham, too, had a shout for a penalty when Steve Davis fell in the box - although replays showed he clipped his own foot rather than any part of Fabregas.
Earlier, Fulham might have made it 2-0 from open play when another error from Lehmann led to the ball squirming out of his grasp. It landed at the feet of Healy but the German managed to scramble the ball clear.
By contrast, Warner could do no wrong and he made another outstanding stop immediately after the break when Rosicky raced clear.
And Arsenal might have been punished seconds afterwards when Simon Davies found himself free in the box but he could only direct a fizzing shot straight at Lehmann.
Still, Fulham - complete with five new faces in their line-up - were on course to clinch a big scalp on their travels and match the total of away victories they had achieved in the previous two seasons, especially as Warner refused to be beaten, this time throwing himself at the feet of Robin Van Persie.
And it appeared to be the same old story for Arsenal as Rosicky missed another chance, this time Zat Knight providing the block.
Then Van Persie whistled a free-kick wide of the post as Fulham survived another assault on their goal.
On came Theo Walcott and, later, Nicklas Bendtner as Arsenal swapped their attacking options.
But you could not help thinking that the one option no longer available to them - Thierry Henry - was just what they were crying out for.
However, a surging run from Kolo Toure was stopped illegally by Carlos Bocanegra and Van Persie belted in the spot-kick to give Arsenal hope.
Remarkably, six minutes later, Hleb twisted and turned in the box before angling a low drive into the bottom corner to hand Arsenal what had looked an unlikely result.
Arsenal boss Arsene Wenger:
"We were strong mentally and physically - this team is resilient. It was a test because we made it difficult for ourselves in the first minute.
"Jens Lehmann is the happiest person in the dressing room now as the team hasn't had to pay for his mistake.
"Fulham played well but we wanted it so much we managed to win the game."
Fulham boss Lawrie Sanchez:
"We feel very hard done by. I felt we were good value for our lead.
"We told David Healy that Arsenal keeper Jens Lehmann can be erratic under pressure and he chased him down and did well to get his goal.
"I was disappointed by the way Arsenal's equaliser came - they had three claims for penalties and then their fourth appeal for a penalty was given."
Arsenal: Lehmann, Sagna, Toure, Gallas, Clichy, Eboue (Walcott 62), Fabregas, Flamini, Rosicky (Bendtner 72), Hleb, Van Persie (Song Billong 90).
Subs Not Used: Almunia, Senderos.
Goals: Van Persie 84 pen, Hleb 90.
Fulham: Warner, Baird, Bocanegra, Knight, Konchesky, Davies, Smertin (Diop 72), Davis, Bouazza (Dempsey 46), Healy (Kamara 63), McBride.
Subs Not Used: Volz, Omozusi.
Booked: Bocanegra, Smertin, Kamara, Warner, Diop.
Goals: Healy 1.
Ref: Philip Dowd (Staffordshire).