Mr Kudrin said he would not serve in a new government next year if, as expected, Prime Minister Vladimir Putin and Mr Medvedev switch roles.
Mr Medvedev gave him a sharp rebuke and ordered him to resign if he continued to disagree over economic policy.
Analysts say Mr Kudrin had hoped to become prime minister himself after December’s parliamentary elections.
A Kremlin spokeswoman said President Medvedev had accepted Mr Kudrin’s resignation.
Mr Medvedev rebuked Mr Kudrin in person on Monday over the comments he made at the weekend in which he said he could not serve under Mr Medvedev if he becomes prime minister because of differences over budgetary policy.
The president reacted angrily, saying the comments were “improper… and can in no way be justified”.
He gave the internationally respected finance minister until the end of the day to resign.
“Nobody has revoked discipline and subordination,” Russian news agencies quoted Mr Medvedev as saying.
“If, Alexei Leonidovich [Kudrin], you disagree with the course of the president, there is only one course of action and you know it: to resign. This is the proposal I make to you.”
“You need to decide quickly what to do and give me an answer today,” Mr Medvedev said.
Mr Kudrin said he would consult with Prime Minister Putin. Later, he offered his resignation and it was quickly accepted.
Mr Kudrin’s resignation will worry foreign investors, who have praised his handling of Russia’s economy, says the BBC’s Steve Rosenberg in Moscow.
He won plaudits for saving much of Russia’s oil revenue in a special fund that helped the country weather the international financial crisis of 2008.
But his relations with Mr Medvedev had been strained for some time, says our correspondent, and he criticised the president’s plans to increase military spending.
Mr Putin’s formal nomination for president is seen as a foregone conclusion.
“I do not see myself in the new government,” he was quoted as saying on Sunday.
“It’s not just that nobody offered me anything. I think that the differences that I have will not allow me to be in this government.”
At a conference of the governing United Russia party on Saturday, Mr Putin announced he would run for the presidency in March and backed Mr Medvedev for the premiership.
He served two terms as president before Mr Medvedev took over in 2008. He was barred by the constitution from running for a third consecutive term.
If Mr Putin wins the presidential election in March as expected, he would be eligible to serve another two full terms, potentially keeping him in office until 2024.
That prospect has alarmed Mikhail Gorbachev, a former leader of the USSR.
“We can assume that there will be no movement forward if there are not serious changes along the lines of a replacement of the entire system,” he wrote in the opposition Novaya Gazeta newspaper, which he partly owns.
“Without this we could lose six years. I think that the future president needs to think about this very seriously.”