They birdied the first extra hole and parred the second. They will resume play at 8:30 a.m. on the 13th tee.
Taylor, the second-round leader, began the day three shots off the lead and shot a 6-under 66 to match Kuchar (69) at 17-under 271.
Rookie Leif Olson (69) tied for third with Tim Petrovic (67) at 16 under. The two shared the first-round lead.
Jimmy Walker (66) and John Senden (67) tied for fifth, another shot back. Third-round co-leader Scott Piercy (73) tied for 12th at 13 under. Defending champion Dustin Johnson tied for 29th at 9 under.
At the first extra hole, the par-5 18th, Taylor hit his third shot from the left rough inside 3 feet for an easy birdie. Kuchar responded by hitting to 9 feet and sinking the birdie putt.
Both found trouble on the next hole, the par-5 12th. Taylor drove way left and his second shot bounded down a cart path near a television tower. Kuchar's second shot also landed near the tower, necessitating a free drop, but both saved par in fading light.
Because the course was still saturated with water after a week of rain, rules officials allowed the players to continue to lift and drop on all areas except teeing grounds, greens and hazards.
Kuchar, who began the day tied with Piercy for the lead, trailed Taylor by a shot after a bogey at 13. Kuchar then tied for the lead with an 18-foot birdie putt at No. 16.
After hitting his second shot to 20 feet of the pin at 17, Kuchar pulled the putt slightly and had to settle for par with just the 624-yard, par-5 finishing hole remaining.
Kuchar then nearly threw the tournament away with one shot and nearly won it with the next. After hitting his third shot into a greenside bunker, he blasted a sand shot that stopped less than 5 inches from the hole.
Kuchar, in the final group with Piercy, got off to a rocky start with bogey on the opening hole when he was unable to get up and down from a greenside bunker. He recovered nicely at No. 2, making an 18-foot birdie putt at one of the most difficult holes at the 7,482-yard Atunyote Golf Club course.
Kuchar also birdied No. 4, then faltered again at the fifth hole. He drove the left rough at the 550-yard, par-5 fifth hole and had to take a penalty stroke, then two-putted from 31 feet for another bogey.
Kuchar rallied with birdie at No. 8 and followed with a clutch 18-foot birdie putt at the ninth hole to tie Taylor at 16 under.
Taylor had two bogeys and four birdies on the front nine and briefly took sole possession of the lead at 16 under when he hit a 70-yard wedge that took one bounce on the green and dropped into the hole for eagle at the par-5 12th hole.
If Kuchar heard the oohs and ahs from the gallery, he was unfazed. He sank an 8-foot birdie putt at No. 10 to break the tie and parred the tricky par-3 11th hole, the most difficult on the course. It yielded only four birdies on Sunday and 27 players, including Olson and Van Pelt, bogeyed it.
Petrovic had three birdies on the front nine to reach 14 under, and birdies at two of the first three holes on the back side tied him with Taylor, one shot behind.
Taylor fell back to 15 under when his second shot at the 15th hole landed in intermediate rough off the back edge of the green and he couldn't salvage par. He rebounded with birdies at the next two holes, tying Kuchar for the lead with a clutch 14-foot putt at No. 17.
Before he teed off at the final hole, Taylor had the lead. Kuchar hit his second shot into the rough to the right of the green at No. 13, barely missing the massive water hazard that comes into play on four holes, and watched in dismay when his 10-foot par putt curled around the cup and failed to drop.
That droped Kuchar into a tie for second with Petrovic and Olson. Taylor then parred the final hole and retreated to the driving range to play the waiting game, wondering if his lead would hold up for his third career victory.
Petrovic had a chance to tie but watched his 16-foot birdie putt at 18 slide by the cup. Moments later, Olson stared in disbelief when his birdie putt at the 17th hole stopped on the lip of the cup and did not drop. He nearly pulled off a storybook finish at 18 when his eagle pitch from the fringe bounced three times and struck the pin but failed to drop.
No rookie has won on the PGA Tour this year, something that hasn't happened since 1998.
Australian Peter Lonard established a course record with a 9-under 63, his only round of the weekend in the 60s.