Noren, chasing a third European Tour title of the season, carded a third-round 65 at The Grove to finish 16 under par, three shots clear of halfway leader Richard Bland.
Peter Hanson, Tommy Fleetwood, Bernd Wiesberger and Richard Sterne were all on 12 under par, with Westwood and Germany’s Marcel Siem a shot further back and McDowell another stroke adrift.
Scot pair Marc Warren and Scott Jamieson are tied in tenth place on nine under par following rounds of 69 and 71, respectively.
Former world No.1 Westwood feels he let down good friend and Ryder Cup captain Darren Clarke after failing to win any of his three matches as Europe suffered their heaviest defeat for 35 years at Hazeltine.
The 43-year-old then missed the cut in last week’s Alfred Dunhill Links Championship, thanks largely to a second round of 82 at Kingsbarns, but carded a flawless 67 yesterday to remain in contention for a 24th European Tour victory.
“You have to be able to forget stuff and move on,” Westwood said. “You can only try your hardest and if that’s not good enough you move on to the next tournament and try to do well from there.
“I played well today and gave myself lots of chances, but you have to stay patient. You look at this course and think, ‘I can shoot really low here’, but if you don’t treat it right and lose patience you can make bogeys quite easily.”
McDowell failed to qualify for the Ryder Cup and was overlooked for a wild card, but made himself watch the action on TV to boost his motivation to make the team in 2018.
“I made myself watch it because I love it, but also because I wanted it to hurt,” McDowell, who won back-to-back French Open titles at 2018 venue Le Golf National, said after a 67 marred by a three-putt bogey on the 18th.
“I needed to watch those guys and realise I want to be one of them again. I still believe I am a top-ten player in the world and I have to start working like that, acting like that and trusting in my game.”
Bland, seeking a first European Tour title at the 395th attempt, quickly doubled his overnight lead thanks to birdies on the first two holes, but dropped his first shot of the week on the fourth after finding sand off the tee.
Noren drew level with a birdie on the sixth and moved in front with another on the tenth, before three more in the last four holes stretched his advantage. Noren was delighted not to drop a shot despite some wayward drives and the 34-year-old’s day was rounded off in style with a birdie from 35 feet on the last.
“I struggled off the tee,” he said. “I was always in the crowd but I thought to myself, ‘If I get it around, keep the putter good, there’s a chance for tomorrow’.
“Obviously on 18 it was an amazing feeling because I thought it was a tough putt, and you can easily three-putt and come off with a sour taste in your mouth. Now it feels great. I’ve three-putted those a lot of times, so sometimes you’ve got to get lucky.”
Bland admitted he had mixed emotions after being unable to hold on to his lead, although two birdies in the last four holes kept him firmly in contention. “I flushed it on the range this morning and started great and then I just kind of lost my way a little bit for nine, ten holes,” he said. “I played good the last five holes to keep myself in touch with Alex and it’s all to play for.
“Sometimes you’ve got to get it done a little ugly and I did that today.”