The South African began the day with a three-shot lead but was caught at the summit at Crans-sur-Sierre Golf Club by a charging Wallace after 16 holes on Sunday.
With both players parring the 17th and 18th to finish on 18 under par, a play-off was required to decide the winner.
And with Wallace unable to save par at the first extra hole, a four was good enough to hand Lawrence victory on his maiden visit to Crans Montana, sparking jubilant scenes on the 18th green.
Lawrence, who was the first winner of the DP World Tour era at the Joburg Open in November after that event was reduced to 36 holes, moves up to ninth on the DP World Tour Rankings.
The 25-year-old made four birdies, a double bogey and a bogey on Sunday to sign for a closing 69.
Wallace, meanwhile, posted a bogey-free 66 to finish two shots ahead of fellow Englishman Richard Mansell in third.
Lawrence began the day on 17 under and picked up an early birdie after holing from 32 feet at the third.
He surrendered a costly double bogey at the fifth after getting into tree trouble off the tee, reducing his lead to one.
A bounce-back gain at the sixth was followed by a stunning hole-out birdie from a tricky spot in the rough around the ninth green, keeping Lawrence one clear at the turn.
Lawrence's 21-foot birdie attempt at the tenth caught the right edge of the hole but did not drop.
He drained his birdie putt from a similar distance at the 12th, though, to give himself a two-shot cushion.
Lawrence got a big slice of luck at the 14th after catching a branch with his second shot but had to settle for a par there after his ten-foot birdie putt just missed.
Playing partner Wallace made a birdie at the same hole to cut the gap at the top to one shot.
And when Lawrence dropped a shot at the 16th, he had Wallace for company at the summit.
A two-shot swing looked possible at the 17th as Lawrence played a sensational bunker shot to set up a birdie chance, while Wallace had to make a good up-and-down to save par.
But with both players making fours there, they were tied heading down the last, where they each made par to take it to extra holes.
Wallace's approach found the very back of the green and he could not get down in two, meaning Lawrence's par was enough to earn the trophy.
Lawrence was thrilled to enter the winner's circle at the 50th staging of the Omega European Masters on the DP World Tour.
He said: "It's a privilege to be able to take this victory.
"There's so much history around this event, all the past champions, so I can't wait to get my hands on that trophy.
"It's a tricky golf course, you can lose a few shots, especially around the 14th, 15th, par fives - you can make a silly mistake and another guy can make birdies.
"But I'm just happy with my patience this week and I'm just so pleased to be able to win this week."
Speaking about moving into the top 100 on the Official World Golf Ranking, Lawrence 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."