Seeking his fourth US Tour title, Simpson coolly birdied four of the last six holes for a five-under-par 65 on the Old White TPC Course to post a 14-under total of 196.
The 26-year-old signed off in style, rolling in a 20-footer on the 18th green to finish two ahead of fellow American Kelly, who rocketed up the leaderboard with a flawless eight-birdie 62.
Long-hitting American JB Holmes carded a 66 to share third place at 11-under with compatriots Ken Duke (65) and US Tour rookie Charlie Beljan (67).
"I just had to be patient there early, I couldn't get anything to drop on the first six holes," world number five Simpson told reporters after finishing with a flawless five-birdie display.
"But this golf course will give you some birdies. If you kind of keep plotting and putting the ball in the fairway, you'll have plenty of wedge opportunities.
"That's what I did, and fortunately for me I took advantage of those holes coming in. It's a great feeling to play another bogey-free round and I'm in great position for tomorrow."
Overnight leader Simpson remained a stroke ahead of the chasing pack when the weather-delayed second round was completed earlier on Saturday but he slipped one behind Kelly after recording a lone birdie on his front nine.
Kelly, who had raced to the turn in a scintillating five-under 29 before picking up his sixth shot of the day at the 12th, went on to birdie 16 and 17 to forge two ahead.
The 33-year-old, a lowly 464th in the world rankings, narrowly missed a long-range birdie attempt at the par-three last but set the clubhouse lead at 12-under before being caught by Simpson who birdied 13 and 14.
Simpson then regained control, getting up and down from a greenside bunker to birdie the par-five 17th before draining the 20-footer at the last.
Asked to outline his strategy going into Sunday's final round, Simpson replied: "I'll see who's around me, but the way I look at it is I've got to go shoot under par tomorrow.
"And if I don't, somebody's going to have to play pretty good. So I will look at it, but it won't really affect the way I play.
"I've almost got to imagine that these guys are all going to shoot four, five under and I've got to stay aggressive," added Simpson, who clinched his first major title by one shot in last month's US Open at the Olympic Club.
Simpson's closest challenger, Kelly, is also the most surprising.
A promising amateur golfer at the University of Washington, Kelly has mainly bounced between the 'mini' tours since turning professional in 2003.
Before this season, his only other lengthy experience of the PGA Tour came in 2009 and his career suffered a major setback when he had to have hip replacement surgery in September 2010 after he was diagnosed with arthritis.
"I didn't know if I could ever get back to a high level," Kelly said. "It took probably about a year to where all the muscle was actually strengthened back up where I could feel normal again."
This season, he has made just seven cuts in 15 starts on the PGA Tour but this week his putting has prospered and he plans to make the most of being in contention going into the final round.
"I've never been in this position before so it will be good for me, it will be a good learning experience for me," Kelly smiled. "I'll just try and do the things I did today and see what happens."