Rose began the day one stroke behind overnight leader Billy Horschel on a tightly-packed leaderboard but birdies at the second and fourth took him to five under.
Rose, who had to qualify this year, is bidding to become the first Englishman to win the Open since Nick Faldo in 1992.
Compatriot Dan Brown, one of six players to start their fourth rounds on three under, was fading though after four bogeys in his opening six holes.
Horschel began in superb fashion with a birdie on the opening hole but bogeyed the third before another birdie at the fourth put him level with Rose and South African playing partner Thriston Lawrence.
After the changeable weather of Saturday which left 24 players within six shots of the lead, the conditions were at least dry on Sunday but with a stiff westerly wind.
Ireland's Shane Lowry, the champion in 2019, suffered a horrible back nine on Saturday and dropped a shot early in his final round, but got into the mix again with four birdies in five holes to get back to four under.
American Russell Henley was on four under after a tidy start featuring four successive pars and a birdie on the par-three fifth, while compatriot Xander Schauffele was also on four under after his first six holes.
World number one Scottie Scheffler was beginning to mount a charge when he got to three under but a bogey at the sixth also left the American three shots back.
Of the earlier starters, Spaniard Jon Rahm made a big move as he went out in 32 strokes to get under par although a bogey at the treacherous 11th hole left him four behind the leaders.
