Links lover Lowry, the 2019 champion, rolled in a superb 20-foot birdie putt on the 18th green to complete a two-under 69 to move to seven under, two shots clear of overnight leader Dan Brown who again impressed with a one-over 72.
Former US Open champion Rose, who had to qualify this year, put together a magnificent three-under 68 in the toughest afternoon conditions to join Brown on five-under.
He went bogey-free for 29 holes and finished by sinking a superb 40ft birdie putt on the 18th green to raise hopes of a first English winner at the Open since Nick Faldo in 1992.
American Scheffler was warming to the task of lifting the Claret Jug for the first time although a bogey at the 18th rather jolted his serene progress on the Ayrshire coast.
Conditions were calmer for the early starters although winds increased considerably throughout the day to stifle scoring and leave the hopes of several big names in tatters, including Rory McIlroy and Tiger Woods.
But Lowry was supreme, making birdies at the first and fourth and after dropping a shot at the fifth he made birdie on the treacherous par-three eighth known as the Postage Stamp.
Majestic finish
His only moment of concern was a double-bogey on the 11th but he quickly shrugged that off and made birdie at the 16th before his majestic finish on the last hole.
"It was pretty good, it was tough out there," Lowry said. "The wind wasn't quite as strong as yesterday but it was still tricky. The 11th was a bit of a disaster but I handled that well and managed to finish nicely."
Scheffler, who has six wins on the PGA Tour this season including The Masters, plotted his way around in impressive fashion with three birdies although he was left frustrated following a bogey five at the 18th after finding a fairway bunker.
"I've played two solid rounds and it put me five shots back, and I'll continue to try to execute and just continue to try to hit good shots and hit good putts out there," he said.
Brown, playing his first major, dropped two shots on his front nine but hung in well and made two birdies on the way back. "I'm proud of what I've done, and hopefully I can continue and sort of still be lingering around on Sunday," he said.
While others lost their heads in the elements, Rose was calm and collected as he showed the virtue of staying out of trouble.
He made two birdies and six pars in his first eight holes, although he finally made a bogey at the 12th. But birdies at the 16th and 18th earned the biggest roars of the day.
"I was aware of the scores around me and knew it was tough so I knew I was doing some good work out there," Rose said.
Several players made use of the relatively benign early conditions including American Billy Horschel and South Africa's Dean Burmester after rounds of 68 and 69 respectively.
American Patrick Cantlay shot a 68 to move to one under, as did Australia's Jason Day, while Canada's Corey Conners (-1) and American PGA champion Xander Schauffele (-1) were the only other players below par after 36 holes.
There was to be no miraculous recovery for Woods though as the American great bowed out with a disappointing round of 77 after carding 79 on Thursday.
The 48-year-old American showed only fleeting flashes of his past brilliance and made just a single birdie.
"It wasn't very good," Woods said. "I was fighting it all day and was never close enough to make birdies."
With the cut projected at plus-six, the likes of former champions Francesco Molinari, Zach Johnson, Henrik Stenson and Louis Oosthuizen also perished, as did McIlroy who despite strong support, never looked like making the cut.
A couple of late birdies were in vain as the world number two ended on 11 over.
US Open champion Bryson DeChambeau, who began the day five over, also endured an early exit on nine over.
