Herbert birdied the second extra hole at Emirates Golf Club to edge out Bezuidenhout after the pair had finished tied on nine under par following matching final rounds of 68.
Defending champion Bryson DeChambeau held a share of the lead after a birdie on the 13th but bogeyed each of the last four holes to finish in a tie for eighth.
Bezuidenhout had looked set for victory on the first play-off hole when Herbert went for the green in two but carved his approach wildly into the water, a shot he described to his caddie as "maybe the the worst shot I've ever hit in my life".
However, Herbert managed to regain his composure and, after a penalty drop, hit a superb pitch to within two feet of the hole to save par before Bezuidenhout was unable to convert a long birdie putt.
Herbert made the most of his reprieve with a huge drive on the second extra hole and this time only needed an iron to find the putting surface to set up a two-putt birdie as Bezuidenhout was unable to get up and down from over the green.
"It's awesome, just the best thing ever," Herbert told Sky Sports. "I've got a bottle of scotch back home in Australia to celebrate so I can't wait to get into that with the boys.
"The last 10 minutes feels like I've just been dreaming. It's so weird.
"Last week I was probably 20th going into the weekend and for about the 10th time in the last 12 months seemed to just back it out and finished at the back of the field.
"I got really frustrated so put in some really good tactics this week with my mental coach Jamie Glazier, trying to be really positive. It's such a cliche but it works so much, I felt so confident out there."
Bezuidenhout had earlier held a two-shot lead as he stood in the 18th fairway in regulation, only to spin his third shot off the green into the water. The resulting bogey opened the door for the chasing pack and Herbert took advantage with birdies on the 17th and 18th to force the play-off.
"I thought Ashun (Wu) was still on 11 under so I tried to make a birdie," Bezuidenhout said. "I laid it up to my perfect yardage, 78 yards to the flag and it just came off a little bit low on the face, a little bit soft."
England's Tom Lewis shared third place on seven under par with Spain's Adri Arnaus and South African Dean Burmester, with overnight leader Wu a shot further back alongside Kurt Kitayama after covering the back nine in 42 in his closing 77.