Douglas Macarthur led for long stretches before Cliffs of Moher (5-1), ridden by Ryan Moore, and Frankie Dettori on the 7-2 favourite Cracksman made their move.
But with those two battling it out, Beggy showed no signs of debutant nerves as he powered Wings of Eagles to victory on the outside.
"I had a great feeling going down," said Beggy, "I probably pulled a little hard, but I got there at just the right time.
"I said I’d follow Ryan (Moore) as it was probably the best horse in the race. I said if I get a run I’ll win. Ryan had gone. I don’t get to sit on too many beasts like this, so I’m going to enjoy it," he added.
The win was O'Brien's sixth in the Derby.
(Reporting by Simon Evans; editing by Ralph Boulton)
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