Shelton struggled when the match resumed after a delay of more than two hours, but came through 6-4, 4-6, 6-4 against his American compatriot in his first hardcourt match ahead of the US Open.
"Happy that I was mentally stable and tough enough to get through that match," said the 23-year-old left-hander, who is seeking his fourth ATP tour title of the year after triumphing in Dallas, Munich and Stuttgart.
Shelton, the world number eight, next faces French left-hander Ugo Humbert for a quarter-final berth.
Shelton led 6-4, 4-3 when play was halted, then saw Damm hold, break on a long forehand and hold again to force a third set.
After trading breaks in the third and fourth games, Shelton broke on a backhand crosscourt volley winner at 5-4 to settle matters after 2hr 9min.
"A tricky match to be in command, in the driver's seat of the match and feel like playing really well, then have a rain delay and the guy come out on fire," Shelton said.
"That was tough to swallow losing that second set the way I did, but happy with the way I fought."
Seven-time Grand Slam singles champion Williams, 46, suffered her 12th consecutive defeat, 6-3, 6-3 to Austria's Anastasia Potapova in a first-round match.
Williams has not won a WTA match since a first-round triumph over American Peyton Stearns in Washington last year.
The Aussie teen, ranked 612th in the world, ousted fellow qualifier Marcos Giron of the United States 6-3, 6-4 in 74 minutes in his ATP main draw debut.
Among his father's 30 career ATP titles was the 2004 Washington crown, four years before Cruz was born.
"It's a pretty special moment," Hewitt said.
"Seeing my dad play this tournament and to get my first tour win, it feels really cool."
Hewitt plays for a quarter-final berth against the winner of a scheduled later match between Greek wildcard Stefanos Tsitsipas and top-seeded defending champion Alex de Minaur, a newlywed from Australia.
Hewitt copied a celebration from his father after the victory, holding up his bent right arm with his hand curled back toward him.
"He just always did it, and when I was little I've just always seen it," Hewitt said. "It's cool to be able to do that now because obviously my dad has been a huge part of where I am today."
Hewitt was uncertain he could copy his father, who won his first ATP match at Adelaide in 1998 on the way to his first tour title.
"I'll try my best," he said. "I'm just playing one match at a time."
Hewitt saw his father's name on the ring of champions at centre court during his qualifying wins.
"It's very surreal," he said. "Pretty special to play my first two qualies matches on centre court and my dad's name being in the back. Kind of inspirational to look up at it as well. Super special to be back here."
Hewitt became familiar with ATP events while being at them with his father but said they have a different and special feel now.
"Trying to forge my own path, use what advice my dad has from his experience, which I think helps a lot," Hewitt said.
"I do feel pretty comfortable being at the tournaments. It's pretty exciting because I've dreamed of playing at these tournaments, and I've seen behind the scenes, and to be in the gym and the locker room with these guys, it's pretty cool."
France's 47th-ranked Adrian Mannarino eliminated American fifth seed Learner Tien 7-5, 6-4 to book a second-round match against American Alex Michelsen, who ousted US lucky loser Mackenzie McDonald 6-4, 7-5.
French eighth seed Arthur Fils was defeated, falling to Spain's 19-year-old Rafael Jodar 7-6 (5), 6-3. The Spaniard reached a second-round match against Japan's 36-year-old Kei Nishikori.
Follow the men's side in Washington here.
Eala downs Zheng to advance
Filipino 21-year-old left-hander Alexandra Eala rallied to defeat China's Zheng Qinwen 4-6, 6-4, 6-1 after two hours and 26 minutes.
Eala, ranked 28th, was down a set and a break but broke six times in all to oust the reigning Olympic champion.
Eala will play for a quarter-final berth against seventh-seeded defending champion Leylah Fernandez of Canada.