Mr Kudrin said he would not serve in a new government next year if, Prime Minister Vladimir Putin and Mr Medvedev switch roles.
Mr Medvedev told him to bug off and ordered him to resign if he continued to disagree over economic policy.
It is reported that Mr Kudrin had hoped to become prime minister himself after December’s parliamentary elections which goes a long way to explaining his hissy fit.
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 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 has been seen as a successful finance minister, both at home and abroad. Foreign investors have praised his handling of Russia’s economy and 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 and he has been critical of the president’s plans to increase military spending.
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 means the world (and Russians) will have to ‘Putin’ up with his Stalin-like personality cult for many years to come. That is just what the world needs. More shots of a topless Putin killing endangered animals.
Speaking of endangered animals what odds do we give of Mr Kudrin dying in mysterious circumstances? The best he can hope for is maybe to get found guilty on fake corruption charges. Of course Vlad the Impaler will have nothing to do with it. He will be as shocked as everyone else.