Northern Ireland's McDowell, who has slumped to 257th in the world rankings, had 15 straight one-putts in a round that included 10 birdies and two bogeys.
His three-round total of 15-under 201 put him one stroke in front of American Chris Stroud in the Dominican Republic tournament, which is played opposite the elite WGC-Match Play Championship in Texas.
"After I knocked the one in on 12 I asked (caddie) Kenny if that was 12 one-putts in a row," said McDowell - and it was.
"I made some putts and I'm going to have to keep doing that tomorrow if I want to win."
McDowell, who won the 2010 US Open at Pebble Beach and has been ranked as high as fourth in the world, won the most recent of his three PGA Tour titles at the 2015 Mayakoba Classic.
Although he has an exemption into this year's US Open at Pebble Beach, he has yet to earn a spot in The Open Championship at his home club of Royal Portrush.
McDowell said he had benefitted from a return to an old relaxation technique.
"I've been working on some routine stuff the past couple weeks and something small has clicked," McDowell said. "It's a breathing thing. It has helped me relax when I'm standing there on the greens.
"Just a little out breath before I take it back. It's something I was doing a long time ago."
But he had little breathing room heading into the final round as Stroud kept the pressure on with a 64 that included six birdies and an eagle at the par-5 fourth.
Stroud would add three birdies in a row at the seventh, eighth and ninth, and two more coming in for a 54-hole total of 202.
It was a further stroke back to South Korean Im Sung-jae and Australian Ian Baddeley.
Overnight leader Im, playing on his 21st birthday in quest of his first PGA title, fired a three-under par 69 for 203 and was tied for third with Baddeley.
Ranked 59th in the world, Im is hoping for a performance that will boost him into the top 50 and into the Masters at Augusta National in two weeks.
Baddeley, seeking a fifth PGA Tour title, signed for a 68.
Leading scores after Saturday's third round of the US PGA Tour Corales Puntacana Resort Championship in Punta Cana, Dominican Republic (USA unless noted, par-72):
201 - Graeme McDowell (NIR) 73-64-64
202 - Chris Stroud 70-68-64
203 - Im Sung-jae (KOR) 67-67-69, Aaron Baddeley (AUS) 68-67-68
205 - Mackenzie Hughes (CAN) 69-70-66, Sepp Straka (AUT) 68-70-67, Kelly Kraft 70-67-68, Paul Dunne (IRL) 66-69-70
206 - Lee Kyoung-hoon (KOR) 71-68-67, George McNeill 70-65-71, Jonas Blixt (SWE) 71-65-70, Jonathan Byrd 68-67-71
207 - Julian Etulain (ARG) 72-69-66, Ben Crane 70-69-68, Brady Schnell 68-70-69, Jhonattan Vegas (VEN) 69-69-69, Ben Silverman (CAN) 70-68-69