The Amateur Champion made all the early headlines on Merseyside as he set the target but Fleetwood - followed by a large and vociferous gallery - came home in 32 to join him at the top.
It looked for most of the afternoon like it would be a two-way tie at the top but Grillo had other ideas and he recovered from two early bogeys with a back nine of 31 to also get to five under.
Spaniard Adrian Otaegui, Frenchman Antoine Rozner and American Brian Harman were a shot off the lead, one clear of Swede Alex Noren, India's Shubhankar Sharma, Scot Michael Stewart, 2009 winner Stewart Cink, U.S. Open champion Wyndham Clark and another American in Max Homa.
Fleetwood was born just up the road from Royal Liverpool in Southport and learned his golf on the links of England's north west coast.
He finished second to Shane Lowry at Royal Portrush in 2019 and had a top five last year at St Andrews, with his three top tens in four starts making him one of the favourites coming into the week.
He made a two-putt birdie on the par-five fifth but gave the shot straight back before putting an approach to ten feet at the seventh to turn in 34.
He then holed a 25-footer on the 11th and put another approach to ten feet on the 14th, starting a hat-trick of gains that also contained an up-and-down on the par-five 15th and a 26-foot putt on the next.
"If you’re not going to enjoy this atmosphere and these experiences then what’s the point?" he said. "Make sure you have the time of your life out there.
“Being this close to home is the coolest thing and I’m so grateful to everyone that cheers me on. I am one of them, one of the guys that's out there. I'm a fan of the game. I'm from this area. Yes I feel at home and to feel that support, it means a lot.
"As first rounds go, that's absolutely the one you wanted, and to get off to a good start feels good."
South African Lamprecht's round was the lowest opening 18 holes by an amateur in a Major since England’s Tom Lewis shot 65 in 2011 at St George’s.
He reeled off three birdies in four holes from the third as he turned in 32 and picked up another shot at the tenth where he holed a 14-foot putt.
His aggressive approach got the better of him at the 387-yard par-four 11th when his drive off the tee found the rough and ended in his first bogey of the day.
However, he chipped in from the front of the 14th and then also birdied the 602-yard 15th to get to five under and, although he bogeyed the next, he was on in two at the last and two-putted for his seventh birdie of the day.
"It's pretty surreal," he said. "It's nice to see a lot of work behind the scenes pay off. It's something I haven't dreamt of yet, but it's pretty cool.
"I mean, as an amateur, yes, it is [surprising]. But in my own head, no, it's not. I think I'm very hard on myself, and I think I earned my spot to be here. I think the way I played today I earned to be on the top of the leaderboard, as of now.
"It's not a cocky thing to say. I just personally think I believe in myself, and I guess stepping on to the first tee box if you're a professional or a competitor, you should be believing that you should be the best standing there.
"Yeah, I'm very proud of it. I'm a little bit surprised, obviously, naturally, but I'm also... I played good golf today."
Argentinian Grillo now plays his golf on the PGA TOUR - where he won the Charles Schwab Challenge in May - but he has 127 DP World Tour appearances on his CV and finished second at the 2014 Dubai Desert Classic.
Bogeys on the first and third put him on the back foot but he two-putted the fifth and put his tee-shot to 15 feet on the ninth to turn in level par.
He then put his approach to six feet at the 11th and holed from off the green at the next before he made the most of the 15th, put a remarkable tee-shot to six feet at the treacherous 17th and holed from 51 feet at the last.
"This is what I'm here for," he said. "I love it. I'm here to play as many holes as I can with the lead and just enjoy it. It's one of the greatest honours in the world and I'm up there. I guess goal achieved for today.
"The four Majors are the things that we want the most. Everybody is here with the same goal. Everybody is here to do the same, play the best and lift the Claret Jug on Sunday. That's kind of the goal."
Otaegui made six birdies and two bogeys in his lowest Major round to par while Rozner took the same route to his lowest opening round in one of golf's big four.
Royal Liverpool member Matthew Jordan had the honour of hitting the opening tee shot and he finished the day at two under alongside 2017 champion Jordan Spieth, Swede Alexander Björk, Italian Guido Migliozzi, England's Oliver Wilson and South Korean Si Woo Kim.