Putin had previously said democracy must be adapted to Russian political conditions.
Medvedev said he chose to speak to Novaya Gazeta because it had never "licked up" to anyone.
Putin, his predecessor as president, has never spoken to the newspaper, which has accused the former KGB spy of crushing freedoms, and is known for investigating corruption and human rights abuses.
Fred Weir, Moscow correspondent for the Christian Science Monitor newspaper, told Al Jazeera that Medvedev's interview was an "extraordinary gesture".
|Novaya Gazeta is known for investigating corruption and human rights abuses [AP]
He said the president, by talking to a "respected and powerful opposition voice", could be attempting to "win points with the liberal intelligenstia who are still a fairly influential group".
"He may ... be trying to differentiate himself from his predecessor who quite pointedly scorned the liberal opposition and Novaya Gazeta.
"So Medvedev is sending a very, very different signal here," he said.
Four of Novaya Gazeta's journalists have met suspicious deaths over the past decade, with critics saying the incidents are connected to Russia's security services.
Anna Politkovskaya, a well-known investigative journalist, was shot dead in Moscow, the capital, in 2006, while Anastasia Baburova, a reporter, was murdered in January this year.
In 2003, Yuri Shchekochikhin, the newspaper's deputy editor, died from an unexplained illness, which his colleagues said was a result of poisoning. Three years before, another reporter was beaten to death.
Medvedev did not discuss the journalists' deaths in his interview, but Natalya Timakova, his spokeswoman, said he wanted to show support for a publication that had suffered so many losses.
The Kremlin and Russia's government have denied any suggestions of a rift between the president and prime minister.
Some analysts have suggested that Medvedev, who has been president for nearly a year, may be starting to assert his authority, but sceptics insist Putin remains in control of Russian politics.