Osaka, who won the first of her two Grand Slams at Flushing Meadows in 2018, wobbled in the second set before recovering for a 6-2 5-7 6-2 victory.
Elsewhere, teenager Coco Gauff’s hopes of another Grand Slam run were ended by 31st seed Anastasija Sevastova.
But top seed Karolina Pliskova and sixth seed Petra Kvitova advanced.
Osaka, 22, withdrew from the Western and Southern Open final on Saturday with a left hamstring injury, saying afterwards she was “stressing” about her fitness going into the Grand Slam.
During Monday’s less-than-convincing win over 81st-ranked Doi, she did not wear any strapping or need medical treatment.
“It was difficult [physically]. I knew there was a chance it would get really long. I will have to see what happens tomorrow and how I feel,” Osaka said.
The former world number one struggled with her first serve in the early part of the match, landing only 38% as Doi took control in the second set and threatened to level the match.
But Osaka switched gears with a decider looming, finding more power and precision with her serves and returns to fight back from 5-2 down to level at 5-5.
Osaka was unable to take any of her three break points for a 6-5 lead before the momentum swung back to Doi, who broke on her second set point through Osaka’s 20th unforced error.
However, Osaka increased her level again at decisive moments. She broke in the opening game of the decider, and again for a 5-2 lead, to come through in two hours and three minutes.
Osaka walked out on court wearing a face mask with the name of Breonna Taylor, a black woman who was shot dead by a policeman in the United States in March.
Asked about it in her on-court interview, she said: “I have seven. It is quite sad seven masks isn’t enough for the amount of names – and hopefully I’ll get to the final and you will see all of them.”
She later added in a news conference: “For me, I just want to spread awareness. I’m aware that tennis is watched all over the world but that maybe there is someone who doesn’t know Breonna Taylor’s story. Maybe they’ll Google it after or something.
“I feel the more people know this story the more interested they will become in it.”
Osaka initially pulled out of her Western and Southern Open semi-final last week as a protest against racial injustice.
Tournament organisers then suspended play in New York on Thursday, one of a number of sporting events in the US to do so in protest at the shooting of Jacob Blake – a black man who was shot seven times in the back by police on 23 August in Kenosha, Wisconsin.