Osaka didn't play at Wimbledon after pulling out of the French Open following the first round for mental health reasons.
It was tied at 4-all in the first set before Osaka's power started to make its mark. The No. 3 seed won the next seven games to build a 5-0 lead in the second set. Osaka also compiled a 34-10 edge in total winners and saved all eight break points Bouzkova earned.
She won her second US Open title last year by beating Victoria Azarenka in three sets.
"I mean, for me I've played a lot of matches [at Arthur Ashe Stadium], I think it might be the court that I've played the most matches in my career," Osaka said. "But yeah, definitely I feel really comfortable here and I think playing a night match is a really big honor of course. Playing the first night match is something I've never done before so yeah I'm just glad I won."
Earlier Monday, rising star Coco Gauff shook off a shaky stretch in her opening match before rallying for a 5-7, 6-3, 6-4 victory over Magda Linette. The No. 21-seeded Gauff advanced to face 2017 US Open champion Sloane Stephens in the second round.
She had hopes of doing it more easily after taking a 5-3 lead and serving for the first set. But Linette won the next six games to take the set and build a 2-0 lead in the second before the 17-year-old American regrouped.
"I just remember, like, last year, not necessarily being bored, but it was just so quiet for the US Open,'' said Gauff, who lost in the first round of last year's tournament. ''I definitely missed the buzz."
Stephens had perhaps the toughest match of them all, needing 2 hours, 10 minutes to edge Madison Keys 6-3, 1-6, 7-6 (7) in a rematch of her victory in the 2017 final.
Arthur Ashe Stadium was far from filled for the start of that match, possibly because fans faced delays entering the Billie Jean King National Tennis Center on the day they were allowed to return to the tournament.
Halep, who had consecutive first-round exits in the 2017 and 2018 US Opens, held off Camila Giorgi 6-4, 7-6 (3). She didn't play in last year's fan-less event.
"So for me it was normal to have the people in the stands and the noise that is always in the US Open,'' Halep said. "I started to like it. I start to embrace these things. I had fun on court, and I liked some people that were screaming in Romanian also and in English.''
Muguruza, the No. 9 seed, advanced with a 7-6 (4), 7-6 (5) victory over Donna Vekic.
The Spaniard, who once was ranked as high as No. 6, announced last September that she would undergo six months of chemotherapy. Having previously planned to retire after last season, she instead returned from her treatment for the French Open, where she lost in the first round. She lost again in the first round at Wimbledon before winning a round at the Olympics.
Suarez Navarro said afterward it was her last singles match and that she was leaving happily.
"I lost, but this year for me was a gift, you know,'' she said. "Last year on these dates, I don't know if I can be here one more time or not, and I'm here.''
Parks hit that mark during a 6-3, 7-5 first-round loss to Olga Danilovic on Court 13 at Flushing Meadows. It was the first Grand Slam main-draw match of the 6-foot-1 Parks' career.
Williams established the record during a first-round match at the 2007 US Open.
For her match Monday night, Osaka entered Ashe with her usual first-round Slam jitters, but not because of any sense of pressure to win, Osaka said afterward.
Instead, it was based on a desire to play well, especially for the loud and appreciative full house that was there a year after she won the title in a nearly empty arena because spectators were banned then amid the coronavirus pandemic.
"I tell people that I'm a perfectionist. I think for me, something that's less than perfection, even though it might be something great, is a disappointment. I don't really think that's a healthy way of thinking, so [that's] something that I really wanted to change," said Osaka, who next faces 145th-ranked Olga Danilovic.
"In this tournament I just want to be happy with knowing that I did my best and knowing that even though I didn't play perfect, I was able to win a match in two sets. ... It's more like a life thing," she continued. "Like, I hope I can keep this mindset throughout my life going forward."
Maybe she wasn't perfect against Bouzkova, who is now 1-11 in Slam matches. After all, the score was 4-all in the early going. From there, though, Osaka reeled off eight of the last nine games.
She was helped by a serve that reached 120 mph.
"Last year, when we didn't have a crowd, I know it felt quite lonely for me," she said with a smile during her on-court interview. "I'm glad to see little kids in the audience -- and, of course, grown-ups, too."
Osaka improved to 55-14 in matches at the majors, including 22-3 at the US Open as she tries to become the first woman with consecutive trophies in New York since Serena Williams collected her third in a row in 2014.
Osaka beat Williams in the chaotic 2018 final in Ashe. She also has won hard-court major titles at the Australian Open in 2019 and this February.
"I've played a lot of matches on this court," Osaka told the Ashe fans. "Definitely, I feel really comfortable here."
In other matches on Monday, three-time US Open runner-up Azarenka beat Tereza Martincova 6-4, 6-0. Azarenka, seeded 18th, trailed 4-1 in the opening set before seizing control.
Halep missed the French Open and Wimbledon -- the two major tournaments she has won -- with a left calf injury, and her return to Grand Slam tennis came at the one where she has had her least success.
Her loss to Kaia Kanepi in 2018 was the first time the No. 1 seed lost in the first round of the US Open in the professional era. She acknowledged after that defeat that maybe the noise of New York just wasn't her scene, and the sounds were back Monday with the stands again allowed to be filled to full capacity.
"I lost a few times in the first round here, so I got a little bit of the experience from those moments,'' Halep said. "There is no other way. So I have to accept it. I have to like it and to be part of it.''
Long lines and big crowds were a readjustment for any player. Keys had to wait while serving midway through the second set of her match against Stephens as fans were late returning to their seats after a changeover.
Halep was in some danger again against Giorgi, an Italian who recently won a hard-court title in Montreal for the biggest victory of her career. The No. 12 seed was broken while serving for the match at 5-4 but recovered to win the final four points of the tiebreaker.
Muguruza has also won the French Open and Wimbledon, but the US Open is the only major where she hasn't reached the quarterfinals. She was knocked out in the first round just two years ago.
But she steadied herself in both tiebreakers against Vekic, who reached the quarterfinals in 2019.