The Thai star defeated China’s Zhou Jingjing 15-8 in the bronze medal match on the second day of competition at the COVID-delayed biennial tournament.
The Chiang Mai native apologised to the Thai fans for her failure to win gold.
“I’d like to thank all Thais for your moral support,” she said.
“I’m sorry that I could not reach my target of winning gold. I did my best.”
The 47-year-old said she was looking forward to the next Games in Paris in 2024.
“I want to go to Paris and I will work harder,” said Saysunee.
She will receive a B3 million bonus from the National Sports Development Fund.
According to the agency’s reward scheme for Paralympians, a champion will get B7.2mn and a silver medallist B4.8mn.
Saysunee won gold medals at the 2004 and 2012 Games and took silver in Rio five years ago.
Thailand aim to claim four to six golds at Tokyo 2020 which ends on Sept 5.
They are pinning their hopes on athletes in wheelchair racing, boccia and table tennis, among other disciplines.
The Kingdom took six gold, six silver and six bronze medals at Rio 2016.
In yesterday’s action, Saysunee completed the morning’s preliminary matches with six wins and two losses, and received a bye in the last 16 round.
She defeated Alesia Makrytskaya of Belarus 15-5 in the quarter-finals but lost to China’s Tan Shumei, 15-9 in the semi-finals.
Tan also beat the Thai in the preliminary round yesterday and in the sabre event on Wednesday.
The gold medal went to Tan, who defeated Viktoria Boykova of the Russian Paralympic Committee 3-0 in the gold medal match.
Meanwhile, the youngest Paralympian in Tokyo took to the pool yesterday looking to change attitudes to disability, as more world records were smashed on the second day of competition.
In a reminder of the continuing risks posed by the pandemic, organisers said a Games participant had been hospitalised after testing positive for COVID-19, but insisted the event would remain safe.
China’s Guo Lingling won the women’s 41kg powerlifting, sending the world record tumbling.
“This is the first time I’ve participated in the Paralympic Games, so to break a world record and win this medal for China is very special for me,” said Guo, 32.
Spanish cyclist Alfonso Cabello also had the best time ever in the men’s C4 1000m time trial.
The Games’ youngest Paralympian, 14-year-old Ugandan swimmer Husnah Kukundakwe, competed in the 100m breaststroke in the SB8 category.
Kukundakwe, who was born without her right forearm and has an impairment to her left hand, did not make the final but said she felt she could “touch the clouds” after a personal best time.
She said she wants to change attitudes towards people with disabilities in Uganda, and give “these kids a chance”.
She is just months younger than fellow teenage swimmer Miyuki Yamada, who became Japan’s youngest ever Paralympic medallist on Wednesday.
Britain’s Paralympians found glory at the Izu Velodrome with Jaco van Gass taking gold and his teammate Finlay Graham winning silver in the men’s C3 3,000m individual pursuit final.
Competitions are taking place mostly without spectators to minimise coronavirus infection risks, and organisers have announced 184 COVID-19 cases linked to the Games.
Tokyo 2020 said that a “stakeholder involved in the Games” has been hospitalised after testing positive for COVID-19.
Japanese media reported that the individual was not an athlete, and does not have severe symptoms.