Rumford lost a two-stroke lead over the final two holes of regulation, with a double bogey at the 17th and a poor drive that left him settling for par on 18. That put him tied with Marcus Fraser and Peter Whiteford at 11-under-par 277 after 72 holes.
Rumford hit a better drive on 18 on the playoff hole and, after Fraser and Whiteford failed to reach the green of the 543-yard hole in two, put his approach 4 feet from the hole for the eagle putt.
It was the first European Tour win for Rumford in nearly six years and fourth victory on the tour overall.
Rumford opened the day two strokes off the lead but, with third-round leader Alexander Noren faltering to a 2-over 74, he closed that quickly. He had birdies on his first two holes and moved in front with four consecutive birdies over Nos. 6-9.
He had bogey at 13 -- for the third straight round -- but got that stroke back at the 14th, leaving him at 13-under. A poor drive on 17 led to a penalty and subsequent double-bogey 6. The closing par gave him a 4-under 68 on the day.
That opened the door for Fraser, who also had a 68 that included eight pars to close the round, and Whiteford, who was 1-over for his final seven holes en route to a final-round 69. Whiteford missed a 4-foot putt on the 18th hole in regulation that would have given him the win.