The burly Lawrence topped a heavily congested leaderboard on seven-under par after nine holes, with England's Justin Rose one shot back and overnight leader Billy Horschel and fellow American Xander Schauffele on five under.
Lawrence is bidding to join a high-profile list of South Africans to have lifted the Claret Jug including Gary Player, Ernie Els and Bobby Locke.
The 27-year-old put together a faultless nine holes that traded four birdies with five pars.
Rose, chasing his second major title and England's first Open victory since Nick Faldo in 1992, hit three birdies in the opening nine holes in relatively benign conditions at Royal Troon on Scotland's west coast.
Horschel, the world number 62, enjoyed the perfect start by rolling in a 23-foot putt on the first for birdie. He dropped a shot at hole three before birdies at four and six.
But a short missed putt on the 100-yard par-three Postage Stamp eighth hole meant he fell behind Rose and Lawrence.
Also in the hunt was Schauffele as he sought to win his second major title in as many months.
The American, fresh from his maiden major win at the PGA Championship, went out in two under to get to five-under par for the championship.
World number one Scottie Scheffler notched three birdies in his first eight holes to put himself in the mix to claim his third major.
But he missed two puts within six feet on the ninth to record a double bogey six and drop to two under.
Ireland's Shane Lowry, the leader after two rounds, quickly went about trying to make amends for a disappointing Saturday, picking up four birdies between holes four and eight to get to four under.
'Straight in the bin'
His two at Postage Stamp was three better than he managed in Saturday's third round, when Lowry slumped to a six-over par 77 as he slid down the leaderboard.
Unheralded Englishman Daniel Brown, ranked 272 in the world and playing his first major, was three over for his opening nine holes to fall out of contention.
Earlier, Spain's Jon Rahm flew out of the traps, birdieing his first three holes to move to one under for the championship but he failed to build on that and was too far back to win.
A number of stars have struggled this week due to the testing weather conditions, thick rough and well-placed punitive pot bunkers.
Rory McIlroy endured two miserable rounds to miss the cut on 11 over, extending his decade-long wait for a fifth major into 2025.
Three-time champion Tiger Woods also missed the weekend, recording his worst-ever performance at the Open with a 14-over score of 156.
US Open champion Bryson DeChambeau, Ludvig Aberg, Wyndham Clark and Viktor Hovland also failed to make the final two rounds.
Home favourite Robert MacIntyre finished nine over after failing to repeat the heroics that secured last week's Scottish Open.
"This week, to be honest, it's just going straight in the bin," said the left-hander.
"Last week is the one that we'll reflect on because there's no real point reflecting on this one when it's been absolute carnage from start to finish."
