The winner came when Macoumba Kandji, a late substitute for forward Omar Cummings, beat Dallas defender Jair Benitez on the right wing, kicking the ball between Benitez's legs. Kandji then crossed the ball into the 6-yard box, where it was deflected into the goal by Dallas defender George John.
The Rapids also had to play a man down the last 13 minutes of overtime.
"I think you've seen just what a spirited group this is," Rapids coach Gary Smith said. "It wasn't necessarily the greatest game of football, but I think we're finding that this team can find a way somehow to win. I'm so proud to be a coach in charge of such a fantastic group of fellows. I'm overjoyed."
David Ferreira, the league MVP, scored for Dallas in the 35th minute. Casey scored the tying goal in the 57th minute.
After a cross by midfielder Jamie Smith, Casey collided with Dallas goalkeeper Kevin Hartman. With both players tangled on the ground, Casey managed to kick the loose ball past Hartman.
"The ball went out to Jamie, he took the guy on, one-on-one, I tried to make a near-post run," Casey said. "I got caught between the keeper and a defender. I was on the ground, under his arm, and no one could get to it but my toe, so I just poked it in."
FC Dallas will be haunted by the sight of the winning goal glancing off John's leg.
"The game of soccer is a cruel game," said FC Dallas coach Schellas Hyndman, the league's Coach of the Year. "You play 90-plus minutes and it's a game of inches. Some days the ball doesn't go in for you."
Kandji was injured on the game-winning goal, and Smith could not replace him because he'd already used the maximum three substitutes. But the Rapids were able to hold on the final 13 minutes for the victory.
Goalkeeper Matt Pickens made a spectacular save late in overtime, and center back Drew Moor kicked a shot off the goal line in the game's waning moments to preserve the victory.
"Matt Pickens, for me, man of the match," said midfielder Brian Mullan, who was acquired in a trade with Houston in September. "He played an incredible match, saved us so many times. I just want to kiss him."
The Rapids were happiest for Pablo Mastroeni, a 13-year veteran of MLS who came to Colorado in 2002.
"I am overjoyed for this guy, to raise that cup and finally sample some success," Smith said. "To get himself a winner's medal is wonderful. I'm so glad I've been here to see it."
Since the beginning of the season, Smith talked about getting Mastroeni some "silverware." Now he has it.
"The feeling hasn't really set in," Mastroeni said. "What has is the resolve that we've shown all year. To finally see it come full circle, and for the guys to be rewarded, for me means everything."
Colorado's victory keeps the MLS Cup in the Intermountain Region; Real Salt Lake won the championship last season, but lost in the first round of this year's playoffs to FC Dallas.
Rapids win in OT
R Colorado wins its first MLS Cup, 2-1, after FC Dallas gives up an own-goal in overtime.• Dallas takes a 1-0 lead in the 35th minute on Martin Chavez's goal.• Colorado's Conor Casey scores the equalizer in the 57th minute.
San Jose 1, New York 1, San Jose advanced on aggregate 3-2
Oct. 30: New York 1, San Jose 0
Nov. 4: San Jose 3, New York 1
Colorado 1, Columbus 1, Colorado advanced penalty kicks 5-4
Oct. 28: Colorado 1, Columbus 0
Nov. 6: Columbus 2, Colorado 1
Eastern Conference Championship
Nov. 13: Colorado 1, San Jose 0
FC Dallas 2, Real Salt Lake 0, FC Dallas advanced on aggregate 3-2
Oct. 30: FC Dallas 2, Real Salt Lake 1
Nov. 6: FC Dallas 1, Real Salt Lake 1
Los Angeles 2, Seattle 0, Los Angeles advanced on aggregate 3-1
Oct. 31: Los Angeles 1, Seattle 0
Nov. 7: Los Angeles 2, Seattle 1
Western Conference Championship
Sunday, Nov. 14: FC Dallas 3, Los Angeles 1
Sunday: Colorado 2, FC Dallas 1, OT