American Kuchar, who clinched one of the biggest victories of his career at the WGC-Accenture Match Play Championship in February, ground out a two-under-par 70 to post an eight-under total of 208 in the weather-delayed US Tour event.
Though every player in the field struggled on lightning-fast greens at Muirfield Village Golf Club as the shifting winds gusted up to 32 mph, ice-cool Kuchar mixed five birdies with three bogeys to take over at the top.
Americans Kevin Chappell (68) and Kyle Stanley (73) were tied for second at six under with Australian Matt Jones (70), 2010 champion Justin Rose of England (71) and second-round leader Bill Haas (76) a further stroke adrift.
"This place is really challenging, particularly with winds at 25 miles an hour all day," Kuchar told reporters after moving a step closer to winning his sixth US Tour title.
"It was a bit of a survival. I was fortunate to make a handful of birdies. Anytime you make a birdie in these conditions, you feel like you're really up on the field here."
Kuchar brought good form into this week, having finished second at the Crowne Plaza Invitational in Fort Worth, Texas, on Sunday, and he oozed confidence after the third round.
"It was a good, solid, steady round of golf," said the 34-year-old. "I drove it well. I feel like I'm hitting it solid, and when you're hitting it solid, the wind doesn't affect the ball as much as the un-solid hits.
"I feel really confident. I had some good stuff going on last week, carrying into this week. It's nice to see it keep going and to have another chance to win late on Sunday will be fun."
The unheralded Chappell, hunting his first US Tour victory in his third full season on the circuit, expressed the views of many when asked how much he had enjoyed the challenging wind conditions.
"I guess it's like a prize fighter," he smiled after his 68 matched fellow American J.J. Henry for the lowest score in the third round. "He enjoys winning, but I don't know if he enjoys getting hit that much."
South African Charl Schwartzel laboured to a 76 to finish at four under. The 2011 Masters champion had birdied his last two holes for a 71 to trim Haas' overnight lead to one shot when the weather-delayed second round was completed earlier in the day.
Danger lurked at every corner of the tree-lined course designed by tournament host Jack Nicklaus and five-times champion Woods struggled as much as anyone in the unpredictable gusts as he carded a 79.
The American world number one recorded a bogey, two doubles and a triple to reach the turn in eight-over 44, his worst nine as a professional, and declined to speak to reporters after bogeying his final hole, the par-four ninth.
"It was a rough day," Woods later told a US Tour official after totalling 30 putts for the third successive round for his worst ever score at Muirfield Village, winding up joint 69th in the field of 72 - a distant 16 strokes off the pace.
"It was tough out there from beginning to end. I tried to fight back on the back nine, just didn't quite materialise."
Former champion Justin Rose had a sense of deja vu after battling through tricky, gusting winds to end Saturday's third round at the Memorial Tournament three strokes off the pace.
Three years ago, the Englishman trailed by four shots going into the final round at Muirfield Village Golf Club and went on to claim his first victory on the PGA Tour by firing a flawless six-under-par 66 in shifting breezes.
"When I won this thing a few years ago, I was chasing on Sunday," Rose said after carding a one-under 71.
"It was similar conditions, and I got it done. This course has a history of the wind swirling. But obviously when it's blowing this hard, it plays a little more consistent. That's the only good thing I can say about it.
"The rest is very tough. The greens are rolling perfectly, but quick, but there's no let-up. You have to hit good shot after good shot, which is how it should be."
Northern Irish world number two Rory McIlroy, who missed the cut here last year, also struggled, mixing two birdies with three bogeys and a double for a 75 to end the day at six over.
The average score in the third round was 73.603, more than a stroke-and-a-half over par, and would undoubtedly have been higher had the greens not been softened by overnight rain.