Settled in mid-division by the 18-year-old, who showed maturity way beyond his years, the Montjeu colt picked up superbly when angled out for a run.
Sea Moon and Ryan Moore had given chase to Brilliant Speed off the turn having missed the break, but St Nicholas Abbey was always going well and swamped them both, stretching clear in tremendous style.
Sea Moon had no answer in second, with Brilliant Speed third.
A clearly moved Aidan O’Brien said: “It’s very special, unbelievable. It was a wonderful ride, it’s incredible. What can I say, he’s an incredible rider. It’s been one of the proudest days of my life, just unbelievable.”
Joseph’s mother, Anne-Marie, said: “I’m so excited, he gave it a super ride. It’s fantastic. My heart was going 100mph, I could hardly stand up I was so nervous. We are so pleased to be here.”
O’Brien junior rode his first winner in May 2009, and his first Group One winner in May this year on Roderic O’Connor in the Irish 2,000 Guineas. He said: “Obviously there was a fair degree of nerves, but I was looking forward to it.
“He jumped good, had a nice position and travelled well. We got the gaps and he picked up well.
“It’s a dream come true. I was very happy with my position, I was very happy with the horse and he picked up when it mattered.
“I’ve been coming to the Breeders’ Cup since I was very small, but to ride here and to ride a winner here is something you can only dream about – it’s out of this world.
“I owe a big thank you to the owners for giving me the opportunity.”
O’Brien has conceded his future will lie over jumps at some point, as he stands 5ft 11in tall. But he said: “I’ll keep riding on the Flat for as long as possible, and take it from there.”
Future plans for St Nicholas Abbey will be made after O’Brien and the Coolmore partners have a meeting.
Aidan O’Brien had earlier landed the Breeders’ Cup Juvenile Turf with Wrote.
Third in the Royal Lodge Stakes at Newmarket in September, Ryan Moore enjoyed the perfect passage on the inside aboard the son of High Chaparral.
Farraaj briefly flattered for Roger Varian, but as soon as Moore switched Wrote for a run the race was as good as over and he won going away.
O’Brien said: “We are delighted, as these races are very hard to win. We always liked him, but he got a bit unbalanced coming down the hill at Newmarket and after that he had a temperature, and had to have a break.”