The low scoring continued as McIlroy holed from four feet on the first, chipped in for birdie on the fourth and two-putted the short fifth for another after driving the green, only to bogey the seventh and eighth.
That was when his caddie Zack Rasego - who carried the bag for Louis Oosthuizen when he won the Claret Jug at St Andrews in 2010 - broke the news to him that he had made history.
"And I was like, 'What are you talking about?'" Grace said.
Until he eased in a three-foot par putt on No. 18 that followed upon a marvel of a lag putt from yonder in the suburbs of the green, Branden Grace did not know a 62 would edit the voluminous record books of men's major golf. "I am swinging the club well right now and feel in control", he said. This beats Grace's best round in a major (64, R3 of the 2015 PGA Championship) and also beats his career-best round on the PGA TOUR (63, R1 of 2016 WGC-HSBC Champions).
Connelly, who is making his Open debut, had already birdied the first when he holed his approach to the second for an eagle two, his ball taking one bounce before plunging into the cup. None of the six players who shot a third-round 63 in a major went on to win the tournament.
Miller, an Open Championship victor at Royal Birkdale as well, was on the broadcast for Grace's 62, and in classic Johnny Miller fashion, added his own set of qualifiers to the record-breaking round, including calling the course setup "really easy". Li rolled in birdies at Nos. Trying to give the guys a chance to go for it. Things on the 7th, they've moved the tee up 30 yards. Add it up: eight birdies, 10 pars and zero chance of ever forgetting such a round.
"I didn't know what was going on 18, I promise you", he told reporters.
The 47-year-old American posted a 77, which left him 10 over after his first-round 73 and heading home for the first time in four outings at Southport. When I hit a bad shot I just told myself to hang in there and try to hole the putt.
Not really, he's just happy if he can stay in solo third.
"You still have to do it out there", said Grace, who said par felt more like 67 than 70 because of the light wind and turf that was on the soft side. "That's good enough for me".