Johnson, who spectacularly eagled the par-five ninth after hitting a four-iron 211 yards to within two feet of the pin, fired a four-under-par 69 in sunny but breezy conditions at the picturesque Kapalua Resort.
He had a chance to seize the outright lead at the par-five last but narrowly missed his birdie attempt from 15 feet to finish at 14-under 205, level with fellow Americans Webb Simpson (68) and Jordan Spieth (69).
Overnight leader Zach Johnson was two ahead of the chasing pack with seven holes to play but bogeyed 14 and 16 on the way to a 74 and fourth place at 12-under.
Dustin Johnson won last year's title, his seventh on the PGA Tour, by three shots in a tournament reduced to three rounds and forced to a rare Tuesday finish because of relentless howling winds at the Kapalua Resort.
He began Sunday's third round three strokes off the pace and launched his timely surge up the leaderboard with his stunning eagle at the ninth.
"I hit a great drive down there right to the end of the fairway and had a good number, 211 (yards) with uphill," Johnson told Golf Channel. "I just hit a great four-iron right at the flag and it ended up being pretty close."
Asked how he would attempt to close the deal in Monday's final round in pursuit of his ninth PGA Tour victory, Johnson replied: "I've got to take advantage of the par-fives.
"Today, I made a birdie and an eagle but I parred the two on the back nine which is poor for me, especially (after) hitting good drives."
Simpson, the 2012 U.S. Open champion, birdied four of the last seven holes to set the clubhouse lead at 14 under.
"That was really good," said the 28-year-old, who clinched his fourth PGA Tour win at the Shriners Hospitals for Children Open in Las Vegas three months ago.
"I feel like I am getting a little more comfortable with what I am working each day... and I was able to capitalise on those holes where you need to make birdie.
"I've just got to stay patient (on Monday). If you don't get it going early, you've got plenty of opportunities on the back (nine)."
Matt Kuchar made the first significant move of the day among the leaders, birdies at the fourth and fifth putting him at 12 under, just a stroke behind the pacesetting Zach Johnson.
However, Johnson picked up a shot with a two-putt birdie at the par-five fifth to edge two ahead, then regained his three-stroke advantage when Kuchar bogeyed the par-three eighth.
Spieth, the 2013 PGA Tour Rookie of the Year, trimmed the deficit to two with birdies at the eighth and ninth before being joined at 12 under by power hitter Dustin Johnson, who eagled the ninth and birdied the 10th.
Zach Johnson squandered a good opportunity to extend his lead when he missed an eight-footer for birdie at the 12th, and Spieth and Simpson closed in at 13 under with respective birdies at the 14th and 15th.
Johnson ran up his first bogey in 39 holes when he overhit the green with his second shot at the driveable par-four 14th and failed to get up and down from the rough to save par.
That briefly dropped him back into a share of the lead before Simpson knocked in a six-footer to birdie the 16th and move one stroke clear.
Zach Johnson also bogeyed the par-four 16th after a wayward tee shot ended up in an unplayable lie, while Dustin Johnson and Spieth, at the 16th and 17th, rolled in 15-footers to join Simpson in a three-way tie at the top.
Masters champion Adam Scott of Australia, the world number two and highest-ranked player in the elite field of 30, fired a 69 to end the day at 10 under, four strokes off the pace.
The Hyundai Tournament of Champions is the opening event of the year on the PGA Tour, but the seventh in the new wrap-around season for 2013-14.