While the Ayrshire links released its grip earlier in the day to allow several players to zoom into contention with sizzling third rounds, the leaders faced heavy rain and a chill wind in what became a battle for survival.
Florida native Horschel, playing in shirt sleeves when everyone else was decked out in waterproofs, carded a two-under 69 to head in to Sunday's climax on five under.
South Africa's Thriston Lawrence and American Sam Burns took advantage of benign conditions early in the day with six-under 65s while Russell Henley shot 66 to move into a large chasing group all on three under.
England's Open debutant Dan Brown battled superbly in the final pairing with overnight leader Shane Lowry and was the outright leader when he birdied the 16th to reach six under.
But the brutal conditions finally caught up with him as he bogeyed the long par-three 17th which he and Lowry both needed a driver to try to reach off the tee.
Qualifier Brown then double-bogeyed the 18th to fall back to three under but will still have high hopes on Sunday.
For Ireland's Lowry, champion in 2019, his round unravelled after leading on eight under after a birdie at the fourth.
With the conditions worsening on the back nine, Lowry bogeyed five of his last eight holes to end on one under.
PGA champion Schauffele also finished on three under after plotting a tidy 69 while former U.S. Open champion Rose, who had to qualify this year, ground out a two-over 73.
SCHEFFLER FRUSTRATIONS
Only nine players finished under par, including Scheffler who suffered frustrations with his putter but produced a sublime birdie at the 17th on his way to a level-par 71.
More wind is forecast on Sunday when the top 14 on the leaderboard will start separated by only five strokes.
West Ham United fan Horschel, who has missed the cut on six of his previous nine visits to the Open, was one of the few players who appeared to revel in the conditions.
He made four birdies in a blemish-free outward nine of 32, and showed remarkable resilience on the way home.
The 37-year-old saved par at the 14th and 16th with stunning bunker shots and resisted the elements until a bogey at the brutal par-four 18th.
"This round in the Open Championship is by far the best I've played in a major. I had to grind out a score coming in," Horschel told reporters.
"I knew it wasn't going to be pretty. I knew it was going to be tough, but I was prepared for what the last nine holes would entail."
Rose, who along with Brown is bidding to become the first English winner of the Open since Nick Faldo in 1992, was almost as impressive.
"I'm delighted to pull up at the end of the day just one back. It felt like survival mode," Rose said.
"The back nine is as brutal as I've played in a tournament for a long time."
It was all so different earlier in the day as Lawrence needed just 30 shots on his outward nine while American former world number one Justin Thomas took 31 before carding a 67 to reach level par.
South Korea's Kim Si-woo provided arguably the day's standout moment when he made a hole in one at the 238-yard par-three 17th -- the first ace at this year's championship.
Horschel is not worried about sleeping.
"I'll probably watch some darts on TV. I'm very English when it comes to sporting events," he said. "I love watching cricket."
