McDowell needs to win here on Sunday to overhaul Stenson at the top of the Race to Dubai and remains in with an outside chance after a third round of 68 on the Earth Course at Jumeirah Golf Estates.
That left the former US Open champion eight under par and five shots behind Stenson, who had seven holes of his round to play.
"I think there is a weak pulse still there," said McDowell, who carded two bogeys, four birdies and an eagle. "Hats off to Henrik, he came here this week and is doing exactly what he needs to do.
"He has been the best player in the world the last six months and he needs to slip up tomorrow and I need to mount a massive challenge. I have not done much better than 68 round here before and I will need to do so tomorrow.
"I have not played my best the last three rounds but I am proud of how I have hung tough. I am still alive and kicking."
Ian Poulter is also in with a chance of preventing Stenson from becoming the first player to win the FedEx Cup and Race to Dubai in the same season, although he needs to win and for Stenson to finish third or worse.
He began the day five shots behind Stenson but had closed the gap to one with a front nine of 33 and birdies at the 10th and 14th, the Ryder Cup star moving to 12 under par and joint second with France's Victor Dubuisson.
Stenson had briefly held a three-shot lead following birdies at the fifth and seventh, but gave the chasing pack hope with a bogey on the 10th.
Low scores remained possible, with former world number one Lee Westwood carding a 65 that threatened to be even better to set the clubhouse target on 11 under.
Westwood was seven under for his first 11 holes but bogeyed the 12th and was able to find just one more birdie after that, although he did save par on the 18th after driving into the water.
"I got in my own way a little bit around the back nine there," Westwood said. "It's a good 65 but seven under through 11, you're looking for a few less than that.
"My swing hasn't been in sync all year, I'm sort of fumbling my way around the golf course and it's something I'm going to work on over the winter."
Dubuisson moved into the lead with birdies on the 13th and 14th and when Stenson responded with his own birdie on the 14th, the Frenchman promptly birdied the 15th as well.
At 15 under par he led by one from Stenson, with Poulter two shots back after a costly three-putt bogey on the par-three 17th.