Arrogate covered the 1 1/8-mile (1810 metres) distance at Gulfstream Park in an unofficial time of 1 minute, 47.61 seconds to secure the $7 million first prize.
Shaman Ghost finished second, with Neolithic battling home third.
California Chrome, meanwhile, faded down the stretch in the final race of a standout career that included victories at the 2014 Kentucky Derby and Preakness Stakes and the 2016 Dubai World Cup.
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It was a disappointing climax to a much anticipated repeat showdown between Arrogate and Chrome, who had placed one-two after a thrilling finish to the Breeders' Cup Classic three months ago at Santa Anita in California.
"Another incredible performance," Smith told NBC Sports about Arrogate, who started from the one hole as a 4-5 favourite, with California Chrome a narrow second choice.
"I was able to use the first turn to our advantage, stayed on the rail ... it was just a matter of getting the trip.
"I looked halfway down the backside and I saw (jockey) Victor (Espinoza) having the nudge on Chrome a little. He had me right where he wanted me but Chrome just didn't fire his race today at all.
"Believe me, that's not the California Chrome I know. He would have been on top of me. I was able to tip out and he (Arrogate) was gone after that. I don’t mean any disrespect to the other good horses, but he got to gear down the last 100 yards."
Chrome, North America's all-time leading money earner and the Horse of the Year in 2014 and 2016, ended his stellar career placed ninth in a 12-runner field, his worst ever finish in his 27th race.
(Reporting by Mark Lamport-Stokes in St Augustine, Florida; Editing by Andrew Both)
