Williams had not beaten a top-20 opponent in four years but used all of her veteran composure as she overcame deficits in both sets and converted six of nine break points.
She initially appeared outmatched at the hard court tournament as Kudermetova broke in the first and fifth games but mounted an extraordinary comeback to win the final five games in a row to claim the first set.
Kudermetova flipped the momentum as she won the next four straight and nearly closed out the second set in the eighth game but instead helped Williams to break with a double fault and a pair of unforced errors.
The Russian missed another opportunity to win the second set when Williams fended off a break point in the ninth game with an unreturnable serve.
The seven-time Grand Slam winner had the momentum as she converted a break point in the 10th game and a demoralised Kudermetova handed her the match with a pair of double faults - out of nine across the entire match - in the final game.
"I think I'm at a better level than I started the year, to be honest," Williams told reporters.
"I was actually worried about the rain delay, because I'm traumatized by Auckland and all those rain delays and getting injured after that, cooling down and changing serves, going in. I'm like 'Oh my God I can't let that happen again to me'."
"My serve is better. My second serve is better. It's really about playing matches. You can't replicate it. Especially after you take a large amount of time off. You have to keep putting yourself in that position to just play. So that's really where I am right now."
Stephens, who along with Cocciaretto was unable to get a full warm-up in due to rain that delayed the start of the match, broke the Italian five times and saved three of five break points en route to sealing the win in 76 minutes.
She will next face French defending champion Caroline Garcia in the second round.
"I felt like when I came out here I was a little bit slow and obviously she didn't get to do her full warm-up, and in our warm-up it was raining at the end," Stephens, world number 38, said in her on-court interview. "So a lot drama but happy to be through after a slow start."
Cocciaretto took control early on and built a 5-2 lead that gave her two chances to serve out the set. But Stephens dialled up the pace and broke to love on each occasion as she used a five-game winning streak to wrap up the opening set.
In the second, Stephens kept her foot on the gas as she consolidated an early break with a hold to love for a 3-0 lead before going on to end the encounter with another service break.
Stephens can expect a tougher test against sixth seed Garcia, who is 4-2 all-time versus the American including a pair of first-round wins at the Cincinnati Open. Stephens won their most recent encounter at Guadalajara in 2022.
Ukrainian Elina Svitolina was forced to withdraw ahead of her anticipated showdown with Caroline Wozniacki due to foot pain she has been experiencing since her run to the semi-finals at Wimbledon last month.
"I am really disappointed to pull out anywhere but I love the fans in Cincy and love playing in front of them which makes it extra hard but I have to take the medical advice I have been given," she wrote on social media.
"I will do my best to rest and recover in time for the US Open now."
Wozniacki, who returned to competitive tennis this month after a more than three-year break to start a family, is now scheduled to play lucky loser Varvara Gracheva in the night session, weather permitting.
