The eighth seed overcame the 35-year-old former Wimbledon finalist – a lucky loser from qualifying – after a marathon first set tiebreak, winning 7-6 (11) 7-5 in two hours and 18 minutes.
Anderson, who made the Wimbledon final in 2018 and the US Open final in 2017, gave his 31-year-old opponent a torrid time on serve, pounding down 13 aces to Evans' five.
Both had a handful of chances to wrap up the first set in the tiebreaker. Anderson led at 6-5, 7-6 and 11-10, but Evans dug his way out of trouble and finally took the set by converting his own fourth set point.
The two men were almost inseparable in most key areas. Anderson won 88% of his first serve points compared with Evans's 80%, while the British number one had an edge on second service points – 62% to 56%.
But Evans gained his advantage by converting one of the two break points he had against Anderson's booming serve, while managing to save all four break points against his own.
The 22nd ranked Evans has played 28th ranked Norrie only once on the ATP's main tour – in the first round of this year's Australian Open – with Norrie winning in four sets.
The pair also met in the final of a challenger series tournament in Aptos, California, in 2016, with Evans winning 6-3 6-4.
The San Diego hard-court tournament was added to the autumn ATP schedule to make up for events that were dropped in Asia due to the pandemic.
British number four Dart was defeated 6-3 6-4 by the Russian – ranked 120 places higher than her in the world at 31 – in an hour and 27 minutes.
Dart had suffered defeat in the final round of qualifying but benefited from a late withdrawal to enter the main draw as a lucky loser.
The 25-year-old subsequently saw off Brazil's Beatriz Haddad Maia in round one.