The two men had been all square after 23 holes when darkness fell on Saturday, and returned to the course early on Sunday.
Sixth seed Casey promptly pared the first hole of the day, the par-four 10th at Dove Mountain, to eliminate Villegas, who hit a poor drive and made bogey.
Villegas was left to rue what might have been, after missing a three-foot putt that would have won the match at the 23rd hole on Saturday.
Casey had no time to celebrate as he had to go straight back out for the 36-hole final against Ian Poulter, who won his semifinal 7&6 against Sergio Garcia of Spain on Saturday
Earlier, Ian Poulter equaled the shortest match of the tournament to reach the final.
The final stroke of a long day belonged to Villegas, a 28-year-old Colombian, who had a 3-foot par putt on the 14th hole to win the match. He pushed it badly to the right, giving Casey an unlikely reprieve.
"I should have made that putt," Villegas said.
On the previous hole, Villegas hit a remarkable bunker shot from 50 yards that stopped 2 feet from the hole for a conceded birdie, only for Casey to knocked in a 6-foot birdie to extend the match.
Villegas also escaped trouble on the par-5 11th when he blasted out of a desert bush, hammered a fairway metal onto the green and halved the hole with a par.
"If I had to get up early, I wanted to be in the final. I didn't want to be continuing a semifinal," said Casey, who reached the championship match last year only to lose to Australian Geoff Ogilvy. "One of us has to be in the final. And both of us want to be there." Some three hours earlier, Poulter closed out Spaniard Sergio Garcia on the 12th hole, 7 and 6.
Poulter was in his room at the Ritz-Carlton, waiting to find out his opponent before taking a hot bath. The match was so long that he wound up taking the bath and then getting a massage. He posted on Twitter, "laying on the massage table having some treatment getting ready for tomorrow, cant believe they are still out there." Both players were exhausted, especially having endured quarterfinal matches Saturday morning, then coping with a few hours of wretched conditions - cold, wind and rain that briefly halted play.
It was only the second time in the 12-year history of this tournament that a semifinal match went into overtime. The other time was in 2004, when Davis Love III defeated Darren Clarke.
Casey is trying to give this World Golf Championship its first All-England final.
Earlier in the day, Casey built an early lead and defeated British Open champion Stewart Cink, 5 and 4. It was the fourth time in as many matches that Casey had closed out his opponent by that score.
Cink's loss meant no Americans reached the semifinals for the first time in tournament history.
The morning chill turned worse as gray clouds moved in, and rain began falling as the quarterfinals were ending.
The rain came down so hard that play was halted for 10 minutes early in the semifinal matches.
The golf was as miserable as the weather.
Garcia took five shots to reach the first green before conceding the hole. Casey topped a tee shot so badly on the 209-yard third hole that it traveled barely 100 yards and didn't even reach the water hazard.
"It was just nice that rain did back off so we could actually play some sensible golf," Poulter said. "It certainly wasn't enjoyable. The golf would have been terrible to watch." Poulter had a blast when the weather improved, winning four straight holes to build a big lead, then effectively closing out the match with four straight birdies.
Poulter won seven holes in an eight-hole stretch, but it was the one he lost - to a par by Garcia - that caused some consternation.
Garcia was 75 feet away for birdie on No. 7, while Poulter went over the green and into the desert, his ball stuck behind a bush. He asked for relief from a television tower and was denied. Poulter protested that he was capable of hitting through the bush and over the tower, a shot he would only consider in match play.
Calling in the chief referee, he won his appeal. But when he realized that his free drop would be in a thicker bush, he opted not to take relief, then played away from the tower. Poulter made bogey, forcing Garcia to nervously knock in a 6-footer for par.
After the match, Garcia was asked if he felt Poulter should have played the shot he intended when asking for relief.
"Well, probably," Garcia said. "That's what I would have done after trying to get relief. But he did what he thought was right, and he's the one who has to live with it, so he'll be fine." That phrase - "he's the one who has to live with it" - can suggest that Poulter did something wrong.
Poulter said he wasn't put off and stood my his decision.
"It's my prerogative," he said. "Do I want to give Sergio the hole if it doesn't come off? I want to see him putt, and I've got every right to see him putt." Poulter had a far more difficult time reaching the semifinals.
Thongchai Jaidee of Thailand, the No. 48 seed, had never trailed in 59 holes until Poulter won the 17th hole with a par to go 1 up. Poulter won on the 18th when Thongchai's 15-foot birdie hung on the lip.
In the other quarterfinal matches, Garcia pulled away late over Oliver Wilson of England, 4 and 3; and Villegas built a 4-up lead at the turn and beat South Africa's Retief Goosen, 4 and 3.
