Williams will open doubles play with her sister Venus on Wednesday.
Williams will face world No. 2 Anett Kontaveit of Estonia in the singles round of 64, also on Wednesday. It will be their first career meeting.
After early struggles with her serve Monday evening, Williams took 10 of the final 13 games against Kovinić, a 27-year-old from Montenegro who is ranked 80th in the world.
After the match, during a ceremony celebrating her tennis career, Williams said she got a boost from the vocal crowd of almost 24,000 people. "The crowd was crazy," she said.
Williams double-faulted twice in the first game but eventually upped her first serve percentage to 66%. She won 33 of 43 first serve points.
A key moment came in the sixth game of the first set when Kovinić appeared on the verge of a 4-2 lead. But a Williams shot that looked long and wide was neither, nipping the back of the baseline and outside edge of the singles sideline. It was the first of 11 straight points for Williams as she coasted the rest of the way.
It was Williams' third match since announcing she will "evolve away from tennis."
"I have never liked the word retirement. It doesn't feel like a modern word to me. I've been thinking of this as a transition, but I want to be sensitive about how I use that word, which means something very specific and important to a community of people," Williams said in a Vogue article published earlier this month.
"Maybe the best word to describe what I'm up to is evolution. I'm here to tell you that I'm evolving away from tennis, toward other things that are important to me," she said.
At her post-match news conference, she was asked if this is definitely her last tournament.
"Yeah, I've been pretty vague about it, right?" she said with a smile. "I'm going to stay vague because you never know."
But earlier on the court, Williams, 40, said it was a difficult decision to move on.
"I think when you're passionate about something and you love something so much, it's always hard to walk away," she said. "Sometimes I think it's hard to walk away than to not. That's been the case for me."
When asked by Gayle King what winning will be for her in the future, Williams pointed to her venture capital company and also said she wants to work on her spiritual life.
Williams' singles win was witnessed by her husband, Alexis Ohanian, and their daughter, Olympia, with the youngster wearing white beads in her hair, reminiscent of Williams' look when she won the first of her six US Open titles as a teenager in 1999.
"I look forward to waking up and just being like, OK, I don't have to run to the court today," Williams said. "I look forward to just being a mom. She's such a good girl. I just want to be a good mom to her."
Also on Monday, Daria Snigur of Ukraine, who won qualifying matches to get into the tournament, upset No. 7 seed Simona Halep of Romania in three sets 6-2, 0-6, 6-4.
"This match for Ukraine, for my family, for all fans who supported me. I want to say thank you all," an emotional Snigur said.
It was Snigur's first-ever Grand Slam match at this level. She won the Wimbledon girls' singles title in 2019.
CNN's Jill Martin contributed to this report.