Shimomura told a post-cabinet meeting news conference on Friday that Abe had asked him to stay on until a cabinet reshuffle planned for next month.
He will return half his pay for the six months through September in atonement, Kyodo news agency said.
"I have caused great worry and trouble to many of the people over the national stadium problem," Shimomura said, adding that Abe had reluctantly accepted his resignation.
Abe announced in July that plans for the stadium, also meant to have been the centrepiece for the 2019 Rugby World Cup, would be scrapped after the estimated cost of UK-based Zaha Hadid Architects' futuristic design ballooned to over $2 billion, nearly twice the original figure.
A third-party panel examining the cost blowout blamed both the Japan Sports Council, a state-run body overseeing the project, and the sports ministry, Japanese media reported.
Japan last month approved guidelines for its new Olympic stadium, vowing to build an athlete-friendly stadium as cheaply as possible and complete it by March 2020, a year later than planned, but without including any cost estimates or limits.
In a further embarrassing blow, Tokyo 2020 Olympics organisers this month scrapped the Games logo over accusations of plagiarism, maintaining that it was not copied but saying too many doubts had been raised for it to be usable.
Abe, who was formally re-elected as ruling Liberal Democratic Party chief on Thursday, plans to reshuffle his cabinet in early October, although key ministers such as Finance Minister Taro Aso and Chief Cabinet Secretary Yoshihide Suga are expected to remain in their posts, media said.
Shimomura, 61, who assumed his post as sports and education minister when Abe returned to office in December 2012, shares Abe's agenda of shedding the shackles of Japan's post-war pacifism and recasting Japan's wartime history in less apologetic tones.
(Reporting by Linda Sieg and Kaori Kaneko; Editing by Chang-Ran Kim and Michael Perry)
