Evans saved four break points in the last game before ending the match with a service winner. At 33, he is the oldest champion in Washington since Jimmy Connors was 35 when he won in 1988.
Evans arrived in Washington with an 8-18 record this season before going 5-0 in this tournament.
"I had to keep telling myself it was match-by-match and not so long ago I wasn't playing great," he said.
"It was a good effort mentally to just stay in the present and keep fighting for each point. And I did a great job of that this week."
Action was suspended on Sunday at 2-all in the second set due to a thunderstorm. After play resumed in front of just a smattering of spectators, Evans needed around 25 minutes to wrap up his first championship at a 500-level tournament, before more rain returned.
"Any title is special. It's obviously the biggest of my career," Evans said. "I played some very good tennis this week and super happy to come out on top."
Evans was seeded No. 9 in the hard-court tune-up for the US Open and will move from No.30 to a career-high 21st in the rankings on Monday.
His only other tour title came in Melbourne, Australia, in 2021.
Evans eliminated No.2 seed Frances Tiafoe in the quarter-finals. No.12 seed Griekspoor, a 27-year-old from the Netherlands, defeated top-seeded Taylor Fritz in the semis.
"Good week," Griekspoor said, "but not good enough today."
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