Lawrence took a three-shot lead into the final round and was still two clear with five holes remaining after recovering from a double bogey on the fifth with birdies on the sixth, ninth and 12th.
Wallace closed the gap with his fourth birdie of the day on the 14th and found himself in a tie for the lead when Lawrence was unable to save par from a greenside bunker on the 16th.
Both men then parred the last two holes to finish tied on 18 under par, but Wallace three-putted the first extra hole - the 18th - from the back of the green to hand Lawrence the title.
"It's a privilege to be able to take this victory," Lawrence, who won the Joburg Open in November after the event was reduced to 36 holes, told Sky Sports.
"There's so much history going around this event, all the past champions, so can't wait to get my hands on that trophy.
"It's a tricky golf course. You can lose a few shots, especially around 14 and 15, par fives (where) you can make a silly mistake and another guy can make birdies.
"I was happy with my patience this week and just so pleased to be able to win."
The victory will take Lawrence from 131st in the world rankings to inside the top 100 and he added: "It's a dream come true.
"Growing up you always want to be in the top 100 in the world and to achieve it is quite emotional. It's awesome."
England's Richard Mansell finished two shots outside the play-off following a final round of 67, with Scotland's Scott Jamieson, France's Antoine Rozner and Spain's Jorge Campillo sharing fourth place on 15 under.