Grace, ranked world No 35, has previously enjoyed four top-five finishes at majors, including a third-place spot at the PGA Championship in 2015 at Whistling Straits but is not a household name outside the golfing fraternity.
Grace took advantage of ideal conditions and a par-70 layout reduced to 7,027 yards to card eight birdies and surge through the field in the 146th Open Championship.
US Open champion Brooks Koepka was two-under through the front nine as he climbed to -5 for the championship. I knew I was obviously playing really well, and making the turn in five-under was pretty special.
Li began the day 12 shots off Spieth's overnight lead, but equalled the lowest final round in major championship history with seven birdies in his last 11 holes at Royal Birkdale.
Branden Grace made history on a low-scoring Saturday as he became the first player ever to shoot a 62 in a major to spark the chase behind Spieth. The South African struggled to hold back the tears after a magnificent round of golf that climaxed with a brilliant up and down from the back of the final green. A birdie on No. 14 presented the chance of the record as it got Grace to 6-under par, but an eccentric lip-out through the left on No. 15 seemed to foil.
China's Li Haotong soared up the British Open leaderboard with a superb last-round 63 on Sunday and at one stage looked to have a chance of becoming the second Asian man to win a major title.
That gave the 28-year-old from Southend a four-under-par total which was matched by Australia's Marc Leishman, who lost a play-off at St Andrews in 2015. "I was so in the zone, playing so well".
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"I'm in a position where it will be very advantageous, everything I've been (through), the good, the bad and everything in the middle", Spieth said. But nobody had ever managed a 62.
There has been one 63 at Royal Birkdale, where Grace just shot his 62.
Grace, whose caddie told him of the record after he holed out at the last, added: "I feel I'm playing well".
"When you get on a run like that, you stop thinking about golf", he said.
"And obviously it makes me feel a lot better after the score I shot today".
It would also put Spieth one leg away from the career Grand Slam of all four majors with just the U.S. PGA to go.
But the 41-year-old Swede, whose rental house was burgled during his first round on Thursday, joined Johnson on three under par after carding six birdies and a solitary bogey in a 65. He was actually unfortunate not to eagle the penultimate hole, and came to the final tee with the pressure on his shoulders.