In the Western Conference, the Seattle Sounders go through to face the LA Galaxy after stunning Real Salt Lake 1-0 in Utah thanks to a superbly struck goal from Honduran Mario Martinez making his first league start.
In New Jersey, a thrilling second half of a clash which was rescheduled after snow on Wednesday saw two red cards and a missed twice-taken penalty by New York striker Kenny Cooper.
DC, who won the tie 2-1 on aggregate after their 1-1 draw at home on Saturday, will now face the Houston Dynamo over two-legs for a place in the title-deciding MLS Cup on December 1.
But the game looked to have turned towards New York when Cooper went down in the area, under challenge from Hamid, resulting in a penalty and red card for the keeper.
The keeper angrily protested that Cooper had leapt over him but there was another twist as the New York forward converted the spot-kick but referee Mark Geiger spotted encroachment by two Red Bulls players and ordered the kick retaken.
To the delight of the travelling DC fans, substitute keeper Joe Willis then dived low to save the retaken penalty and keep the scores level.
The teams were both down to 10 men after New York's Mexican international defender Rafa Marquez was sent off in the 75th minute when a rash challenge earned him a second yellow card.
Extra-time beckoned but with two minutes of regulation time remaining, DC broke and overlapping full-back Robbie Russell slipped the ball to rookie DeLeon who coolly converted.
United head coach Ben Olsen praised the spirit of his unfancied side, who have progressed without leading player Dwayne De Rosario, who is injured.
"When the chips are down there is a certain spirit about them, it is nothing to do with me, these guys have just come together by themselves," said Olsen.
"They gelled and they believed and sometimes belief is enough," he added.
It was a less dramatic game in Utah with a strong wind having an impact on the game but the Seattle Sounders overcame the 2009 champions thanks to a little magic from Martinez.
Martinez had hit the bar with a powerful drive in the 31st minute while after the interval Sounders' Brad Evans forced a great save out of Salt Lake keeper Nick Rimando.
The breakthrough came in the 81st minute when Colombian Fredy Montero slipped the ball out left to Martinez who blasted it, first-time, into the far left corner.
The win sends the Sounders into the Western Conference final for the first time after three successive defeats at the semi-final stage.