Dmitry Medvedev, who will succeed Putin, was also at the meeting.
Afterwards, speaking aboard Air Force One, the aircraft of the US president, Stephen Hadley, Bush's national security adviser, said that the US and Russian leaders could leave the missile-defence issue to their prospective successors.
Suggesting that Putin would probably stay involved on missile defence and other foreign policy matters, Hadley said: "That seems to be a good thing, not a bad thing".
Russia is also opposed to the eastward expansion of Nato and Washington's support for Kosovo's independence from Serbia, a long-term ally of Moscow.
Putin said that he felt "cautious optimism" that Russia would eventually reach agreement with the US over the missile shield, which Washington insists is not aimed at Russia.
And he said that Medvedev, who is set to become Russia's president in May, would represent Moscow on foreign policy at international meetings.
Bush said that he wanted to "to work through common problems" with Medvedev once he takes power.
For his part, Medvedev told Bush that he hoped to further develop bilateral relations "without interruption", Russian news agencies reported.
"Russian-American relations are a key factor in world security," Medvedev told Bush, Interfax news agency quoted him as saying.
"I would like to act so that our relations develop further without interruption."