Reed made bogey at the first playoff hole to drop out of a three-man contest that also included Scotland's Calum Hill.
The 24-year-old Schott clinched victory on the second playoff hole after Hill hooked his drive out of bounds, shanked his fourth shot into water and shook hands with his German rival, who was on the green in three shots.
"Extremely happy, surprised. I don't know what's happening right now," said Schott, who was clearly battling nerves as he attempted to end his title drought five years after turning pro.
Reed was seeking a second straight win, after the Dubai Desert Classic last Sunday, to complete a whirlwind week in which he also announced he was leaving LIV Golf with the aim of returning to the PGA Tour.
Ten shots off the lead heading into the weekend, the American shot a six-under 66 on Saturday and 67 on Sunday to close on 17-under par.
He was tied for the lead in the final round after picking up a shot at No.14 for a third straight birdie, but played the final four holes in one over.
Schott (69) bogeyed the 17th and Hill (71), the leader of the second and third rounds, three-putted for bogey at 18 as they joined Reed in a playoff watched by Bahrain Crown Prince Salman bin Hamad Al Khalifa at Royal Golf Club.
The leading Australian in the field was Jason Scrivener, who finished with a 71 to tie for 16th on 11 under, while Anthony Quayle finished a couple of shots further back following a final-round 72.
