The American (32) closed with a three-under 69 at Gary Player Country Club in Sun City to edge out young South African talent Aldrich Potgieter.
Veerman began the final round five shots behind the 2022 Amateur Championship winner, making six birdies and three bogeys to set the target at five-under.
Potgieter (20) was seven-under starting the day, but while he triple-bogeyed the 12th to slip to five-under, he birdied the 14th only to bogey the par-three 16th.
The South African was still tied for the lead with Veerman playing the difficult par-four 18th but missed the green right and failed to get up and down for par, missing from 20 feet to force a play-off.
“It’s surreal,” said Veerman, who ended a three-year wait for his second DP World Tour win.
“I can now put on my Instagram that I’m the champion of ‘Africa’s Major’ – the fifth Major.”
He added: “I didn’t have the best seasons the last two years. I put a lot of work in, me and my team – countless hours.
“To win this season, to start off this season with a win is indescribable.
“I knew I had a chance but actually thought I might have lost it on the 16th when I three-putted and missed that putt. That one really hurt but it doesn’t even matter!”
Veerman faced an anxious 30-minute wait for Potgieter to finish and expected extra holes, after his bogey at the 16th.
“I was preparing for a play-off,” he confessed. “I really was in the mindset I was going to go back out there and play again. When Aldrich made bogey on 18, I was almost thunderstruck.
“Aldrich is a world-class player and is going to do amazing things in his career. But the closing stretch of holes is so difficult out here. I knew if I could just par those, I might have a chance.”
As for his relief at winning, he explained the family sold a business his wife ran to join him on the road with their two children.
“It means so much to me, my family, my team,” he said. “It was a lot of work to say goodbye to friends and family for a whole year, so to essentially have it pay off is relieving, satisfying, all of those words.”
Potgieter signed for a 75 and shared second place with Matthew Jordan (72) and Romain Langasque (71) on four-under par as Holywood’s Tom McKibbin finished 60th in the 66-man field on 15-over after a closing 74.
At the Legends Tour’s MCB Tour Championship Mauritius, defending champion Peter Baker finished birdie-birdie to force a play-off, then birdied the par-five 18th a further three times to outlast Simon Griffiths on the Legends Course at Constance Belle Mare Plage.
He closed with a four-under 68 to match Barry Lane Rookie of the Year Griffiths, who shot 65 to set the target at 16-under.
Kilkenny’s Gary Murphy closed with a 72 to tie for 22nd on five-under, finishing 46th in the season-long rankings behind winner Adilson da Silva of Brazil to secure a spot in next year’s Staysure PGA Seniors Championship.
