McIlroy, who had started the round two off the pace, fired a five-under-par 67 on a relatively calm morning at Kiawah Island to post a seven-under total of 209.
Sweden's Carl Petterson, continuing his consistent form throughout this week, shot an even-par 72 to trail McIlroy by three but the fourth round later on Sunday has no shortage of contenders with 14 players within six shots of the lead.
American Bo Van Pelt (67), South African Trevor Immelman (70) and Australian Adam Scott (70) are all well poised in a three-way tie at three under.
But McIlroy, looking confident in what has been his best effort in the majors this season, was delighted to have picked up where he left off before Saturday's storm halted the action.
"I thought it was just a continuation of how I played on Saturday afternoon," McIlroy, one of 26 players who had to complete the third round on Sunday, told reporters after a seven-birdie display.
"I struck the ball beautifully from tee to green, same thing on Thursday, as well. Just one more round like that, and I'll be happy."
McIlroy shot 32 on the front nine before Saturday's downpour brought a premature end to play and after returning for a 7.45 am (1145 GMT) start on Sunday he failed to capitalise on early scoring opportunities.
The Northern Irishman, who shared the lead with Fijian Vijay Singh overnight, squandered three birdie chances and then missed an eight-foot putt to bogey the 13th.
But the 2011 U.S. Open winner bounced back with birdies on the 15th and 16th thanks to putts of 15 and 10 feet.
"I had six feet for birdie on 10, four feet for birdie on 11, 12 feet for birdie on 12, I kept it going and I was giving myself plenty of chances and I was glad I made a couple," he said.
Woods, who made three bogeys from his opening seven holes on Saturday, got off to a bad start with his eight-foot putt for par on the short eighth lipping out for a bogey.
But three birdies on the back nine put Woods, still lacking precision off the tee, back in the frame, though he will be disappointed to have bogeyed the par-three 17th where he found sand off the tee and failed to get up-and-down.
Aiming for his first major title in over four years, Woods was glad to be in the frame.
"After getting off to just a horrific start and then battling back like I did this morning, I gave myself a chance. That's what I wanted to do, at least have a shot going into the afternoon," said Woods.
The 14-times major winner suffered a painful incident on the 15th hole where he was entangled in cactus thorns after a wayward tee shot and came out of the sand dune area limping.
"It itched like hell for about a hole, and then it was gone," said Woods.
Singh dropped four strokes on the back nine, including a damaging double-bogey on the par-four 15th where he drove wide right of the green and then made a hash of his attempted chip.
PGA Championship winner in 1998 and 2004, Singh carded a 74 to end the round five strokes behind McIlroy.