The Belgian overcame the power problems that dogged him on Friday to head into the final day in seventh place, 7min 43.7sec behind rally leader and title rival Ott Tanak.
Neuville went into the decisive weekend with a 25-point lead over Hyundai team-mate Tanak and can clinch the title if he avoids any major mishaps on Sunday.
"I have had enough setbacks throughout my career so I learned to stay calm and just deal with it," said the 36-year-old Neuville, a five-time world championship runner-up.
"The best thing that you can do in that situation is to fight back, and I think we have done that in a proper way today."
Estonia's Tanak, the 2019 world champion, cannot afford to put a foot wrong if he is to stand any chance of snatching the silverware.
He finished the day with a 38.0sec lead over second-place Elfyn Evans of Britain.
"We need to keep going," Tanak said after the 13th stage.
"Japan is never comfortable."
Neuville finished Friday in 15th place after his car lost power in the morning session and his mechanics were unable to fix the problem for the afternoon.
The bespectacled Belgian wasted little time making up lost ground after an overnight service, winning two of Saturday's six stages.
He jumped ahead of Bulgaria's Nikolay Gryazin to take seventh place after a furious late afternoon push.
"We are still within a fight and we still need to catch some points so I'm not thinking too much about it," Neuville said of his title chances.
"I'm sure after a few stages tomorrow we will see a bit more clear where we are and what we have to do."
Neuville's recovery also helped Hyundai's chances of winning the manufacturers' championship.
They went into the weekend with a 15-point lead over Toyota.
The 12th stage of the rally was cancelled for security reasons after an unauthorised vehicle appeared on the course.
TV pictures showed a van blocking Evans' path as he waited on the start line.
Standings: 1. Ott Tanak-Martin Jarveoja (EST/Hyundai) 2hr 39min 48.0sec, 2. Evans-Scott Martin (GBR/Toyota) at 38.0sec, 3. Sebastien Ogier-Vincent Landais (FRA/Toyota) 2:10.9, 4. Adrien Fourmaux-Alexandre Coria (FRA/Ford) 2:19.1, 5. Takamoto Katsuta-Aaron Johnston (JPN-IRE/Toyota) 2:25.2, 6. Gregoire Munster-Louis Louka (LUX-BEL/Ford) 3:07.1, 7. Thierry Neuville-Martijn Wydaeghe (BEL/Hyundai) 7:43.7
Stage winners: Tanak (SS5, SS6, SS13, SS16), Evans (SS4, SS7, SS9, SS10), Neuville (SS2, SS11, SS14), Ogier (SS3, SS15), Adrien Fourmaux (SS1), Katsuta (SS8)
