Bush described relations as candid and constructive
Speaking after their meeting at the Asia-Pacific Economic Forum (Apec) in Shanghai, President Bush repeatedly thanked President Putin for his support over the terrorist attacks on America.
He said it underscored the fact that the US and Russia no longer view each other as enemies.
But he reiterated that the Anti-Ballistic Missile (ABM) Treaty, signed by Russia and the US in 1972, was anachronistic, despite Russia's insistence that it underpins global security.
No Cold War
President Bush noted that Russia was the first to call after the attacks on 11 September.
He said the two countries had made progress in "a new relationship ... based on co-operation and mutual interests instead of confrontation and mutual vulnerability."
Mr Bush said he discussed "significantly lowering" nuclear arsenals with President Putin.
But he also said the terrorist attacks of last month made it clearer than ever that the ABM is "outmoded and dangerous".
The ABM treaty bars President Bush's plans for a US missile defence system, which Washington says will protect it from attacks by rogue states.
Russia opposes both the US missile defence plan and the abolition of the treaty, which it regards as the cornerstone to global security.
But the Russian president told a joint news conference that they had made progress on the issue.
Further talks planned
"At least we do have an understanding that we can reach some agreement taking into account the national interests" of the two nations, Mr Putin said.
A senior White House official said after the news conference that although President Bush did not give President Putin a deadline for the US's exit from the ABM treaty he made it clear that it would happen soon.
This would effectively sound the death knell for the treaty, as only Russia and the US subscribe to it.
Mr Putin said he looked forward to longer negotiations on nuclear stockpiles and missile defence at their meetings at Bush's Texas ranch next month.