Takaichi appointed Yana in a cabinet reshuffle last month but had openly chided him in parliament for the remarks over the road funding, which local media first reported earlier this month.
Yana offered his resignation to avoid causing any more trouble for parliamentary deliberation and diverting attention from overseeing agricultural, forestry and fishery policies, the prime minister said.
"I am the one who made the appointment. Responsibility for appointing him (to the post) rests upon me, the prime minister," Takaichi told reporters.
Yana's replacement will be announced on Thursday, she said.
The resignation comes during an extraordinary parliamentary session in which Takaichi faces a high-stakes challenge to win approval for key policy goals including a temporary sales tax cut on food and beverages.
Opposition parties in parliament's upper house, where the ruling coalition lacks a majority, were preparing a censure motion against Yana while some threatened to block debate on the sales tax legislation until he stepped down, according to local media reports.
Yana confirmed the authenticity of an audio clip, obtained by several local media, in which he said one reason for the budget cuts in two municipalities in his constituency was their support of a rival candidate in a parliamentary election held in February.
He has since denied having the power to sway such budget decisions.
His comment quickly drew widespread criticism as an abuse of power.
An opinion poll by public broadcaster NHK this week indicated 78 per cent of respondents thought Yana's explanation was unacceptable.
Support for Takaichi's cabinet fell two percentage points from last month's poll but remained elevated at 51 per cent, according to the same survey.
Grilled in parliament, Yana has also admitted that his campaign provided free transportation for supporters to attend his election rally, a move that some MPs criticised as a potential violation of election law.
