A few hours later, Mr Medvedev and Mr Putin appeared alongside each other on national television to say the government was stepping down.
"The president would be obliged to appoint them [the parliament's choices for prime minister and other cabinet posts] to these jobs", Mr. Putin said.
Sergey Utkin, the head of strategic analysis at the Moscow-based Primakov Institute of World Economy and International Relations, said that it wasn't yet clear whether Mr. Putin meant to return to the Prime Minister's job again in 2024.
The announcement came as a surprise, and in a statement Mr Putin thanked the government for what it had achieved.
Under the current constitution, which sets a maximum of two successive terms, Putin is barred from immediately running again, but his supporters find it hard to imagine Russian political life without him. Medvedev, a protege who temporarily replaced Putin as president, extended the length of Russian presidential terms to six years from four.
"The main result of Putin's speech: what idiots (and/or crooks) are all those who said that Putin would leave in 2024", grumbled frequently-jailed Putin gadfly Alexei Navalny.
Medvedev, a longtime Putin ally, served as Russia's prime minister since 2012.
He said that the president had "outlined a number of fundamental changes to the constitution", and that "in this context, it's obvious that we, as the government. should provide the president of our country with the opportunity to make all the decisions necessary for this".
In his address, Putin said the constitution must also specify the authority of the State Council consisting of regional governors and top federal officials.
In televised remarks, Mr Putin said Mr Medvedev would take up a new position as a deputy head of the presidential Security Council. Mishustin, who will be quizzed by parliament on Thursday, has played ice hockey with Putin but has little public profile and had not been spoken of as a possible candidate. Putin also proposed that the State Council, now an advisory group to the president, be constitutionally enshrined with formal authority.
"I think all this is a response to opinion polls reflecting popular dissatisfaction with government and their lots in life, and ebbing support even for Putin".
Wednesday's changes will be seen by many as the start of Putin's preparations for his own political future when he leaves the presidency in 2024.
Speculation has swirled about changes to Russia's political system that would allow Putin to stay on after 2024.
Putin seeks to add provisions that would allow him to remain in power indefinitely, although he might still have to forfeit the title of "president".
"Such a model resembling the Chinese one would allow Putin to stay at the helm indefinitely while encouraging rivalry between potential successors", Rogov observed. More Putin in Russian Federation means more Putin on the global stage.
The announcement was made by Prime Minister Dmitry Medvedev, reports suggest.