Westwood had seemingly done the hard work by getting through 16 holes in level par for the day and four under overall, despite playing the vast majority of those in heavy rain at Oak Hill.
But after the rain stopped, the 40-year-old Englishman dropped three shots in his last two holes to card a 73 that left him one under par, six behind Masters champion Adam Scott and five adrift of US Open winner Rose.
Scott added a 68 to his opening 65 and playing partner Rose shot a stunning 66 thanks to a brilliant homeward nine of 29, eclipsing former US Open champion Webb Simpson's course record-equalling 64.
"I played pretty well," said Westwood, who is still seeking his first major title at the 63rd attempt after letting a two-shot lead slip going into the final round of the Open at Muirfield last month.
"The course was playing really long, the par-four 17th was unreachable, and I was just ticking along before getting kicked where it hurts on the last two holes.
"I would like to have finished par, par but I am in red figures going into the weekend and still have a chance."
Rose admitted he had been "hanging on for dear life" after starting on the 10th and dropping shots at his first two holes, but after reaching the turn in 37 the Englishman stormed home in 29 with six birdies - the longest coming from 15ft.
"I sit here today really relishing the opportunity on the weekend to try and win another major with no hesitation, which there may have been a few years ago because you don't know how it's going to pan out or how you're going to deal with it," said Rose, who finished fourth in the Open as a 17-year-old amateur in 1998 but then missed 21 cuts in a row after turning professional the next day.
"It's wonderful to be in this situation right now, talking about having done it, talking about feeling like you can win more, believing in yourself and not talking about how I hope it could happen this week. So I think that alone makes it easier."
Only golfing legends Gene Sarazen (1922), Ben Hogan (1948), Jack Nicklaus (1980) and Tiger Woods (2000) have won the US Open and US PGA in the same year, while Nicklaus was also the last man, in 1975, to win the Masters and US PGA in the same season.
That is Scott's aim after he claimed his first major title at Augusta National in April, the Australian having also finished third in the Open at Muirfield last month.
"I think the platform has never been better for me to go on and win multiple majors," said Scott, who sent his good friend Rose a text saying 'This is our time' after his Masters triumph.
"I guess you've got to take the confidence and form of winning a major and run with it. I've sat in front of you guys and told you that these are going to be my best years, and generally they are for any golfer.
"But I'm doing everything I can to make sure that they are and, you know, I can't take my foot off the gas just because I achieved something great at Augusta."
The course record of 64 was set by Hogan in 1942 and equalled by Curtis Strange in 1989, but Simpson could have created history with the first sub-63 round in major history.
The 2012 US Open champion was seven under par after 15 holes and needed to play the last three in one under to shoot a 62, but could only manage a bogey and two pars.
"It was in my head," Simpson said. "I didn't want to talk about it because I thought it would be the wrong thing to focus on. This game is so funny. When you try to make birdies it seems like you don't. It's so hard because you want to for the record but you can't do that on a golf course this hard."
Defending champion Rory McIlroy, who won five times last year but has recorded just one top-three finish in 2013 since his controversial equipment change in January, was in danger of missing the cut before recovering to card a 71 to lie level par.
"I was letting the round get away from me somewhat, but making four birdies on the last eight holes was nice to sort of redeem the round a little bit and keep myself in the tournament," McIlroy said.
"It makes me feel good because maybe in the middle of the season or a couple of months ago I wouldn't have been standing up here. I would have been going home. It's good to be able to do that and fight back and makes you feel good about yourself going into the weekend."
Scott was initially overtaken by Robert Garrigus, who birdied five of his first eight holes to reach eight under, and when Garrigus dropped back with consecutive bogeys it was fellow American Jason Dufner who moved into the lead.
Dufner went to the turn in 31 and added birdies at the 11th and 13th to be six under for his round and eight under for the tournament.
Dufner also birdied the 16th to move seven under for his round and needing to play the final two holes in one under to shoot the first sub-63 round in major history.
The par-four 17th, which was out of reach with a driver and three wood for most players in the morning rain, did not look like providing a good chance but Dufner was agonisingly close to holing from 20ft after a superb approach.