The watch was given to him by his wife Joanne Woodward as they filmed the 1969 movie Winning together.
Newman, who died in 2008 aged 83, later used it to time car races in which he took part.
The stainless steel watch was sold to an unnamed telephone buyer for $17.8m (£13.6m) – much more than the $1m it had initially been expected to fetch.
It is a record for a wristwatch sold at auction, but a Patek Philippe pocket watch sold for more than $24m in 2014.
In Winning, Newman played a racing car driver, a role which spurred his interest in the sport. He went on to take part in the 24-hour Le Mans race.
Before giving him the watch, Woodward had it engraved with the words “Drive Carefully, Me.” He had been injured in a motorbike accident in 1965.
Woodward bought him a new watch in 1984, and he gave the Daytona Rolex to his daughter’s then-boyfriend, James Cox.
Cox put the watch up for auction. Some of the proceeds will go towards the Nell Newman Foundation, a charity set up by Paul Newman’s daughter, and Newman’s Own Foundation.
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