"This is not an American battle, this is a NATO mission as well, and we are working actively and diligently to consult with NATO at every step of the way," Mr. Obama said.
Added Rasmussen: "Our operation in Afghanistan is not America's responsibility or burden alone. It is, and will remain, a team effort."
Rasmussen said he agrees with the president's approach of devising the proper strategy before considering resources. "The first thing is not numbers," he said, stressing the importance of finding the right approach to the eight-year-old conflict. He said careful decision-making "should not be misinterpreted as lack of resolve."
Rasmussen noted, however, that NATO countries are now looking at the report from Gen. Stanley McChrystal, the commander of U.S and NATO troops in Afghanistan, which calls for troop increases. And he said, "we will stay in Afghanistan as long as it takes to finish our job."
CBSNews.com Special Report: Afghanistan
The president, who lauded Rasmussen as "active and effective," said the two men had a "very fruitful" discussion in which they agreed that it is "absolutely critical that we are successful in dismantling, disrupting and destroying the al Qaeda network."
Mr. Obama is now considering whether to increase troop levels in Afghanistan or shift to a strategy more focused on disrupting al Qaeda in Pakistan, which would rely less on troops.
The president and Rasmussen told reporters they also discussed missile defense and agreed that the new configuration is an improvement. The pair did not take questions or discuss Iran or the McChrystal request specifically.
60 Minutes: McChrystal's Frank Talk on Afghanistan