Westwood looked to have kept his chances alive despite playing the majority of his second round in heavy rain, only to drop three shots in his last two holes to finish one under.
That left the 40-year-old former world number one eight shots off the lead held by Jason Dufner - who equalled the major record with a 63 - but also five adrift of US Open winner Rose and six behind Masters champion Scott.
Scott added a 68 to his opening 65 to finish one ahead of Sweden's Henrik Stenson and playing partner Rose, who shot a 66 thanks to a brilliant homeward nine of 29.
Rose admitted he had been "hanging on for dear life" after starting in the rain on the 10th and dropping shots at his first two holes, but after reaching the turn in 37 he stormed home with six birdies - the longest coming from 15 feet.
"I sit here 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 greats 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. I was hungry before the Masters and I might even have a bigger appetite after it. It might be greedy, but I feel like this is my time to get everything I want out of my career and I'm going to keep pushing until I do.
"I've got to take advantage of it, otherwise it's all a waste."