The senator, who is a key Republican on the Armed Services Committee, had just returned from a trip to Afghanistan to fulfill his army reserve duty.
"After this trip I think we can transition, next summer, some areas of Afghanistan to Afghan control," Graham said on CBS' "Face the Nation" Sunday. "I've seen progress I have not seen before, so I see a scenario if things continue to develop the way they are, certain areas of Afghanistan can be transitioned to Afghan control and we can remove some troops safely without undermining the overall war mission."
"This is a change in position for you," host Bob Schieffer said.
"Yes," Graham answered.
"Six weeks ago you were not saying you thought we could safely begin to withdraw. You are now saying things have changed and you think that's possible," Schieffer said.
"It's due to progress I've seen on the ground. It has to continue," Graham said.
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Graham capped his newly optimistic view of the war effort with a sports analogy.
"We are on offense now, we have got the ball back. For years we have been playing defense. By next summer I think we can cross midfield and some of our troops can come home. But having said that, there will be a substantial need for many troops there well after July 2011."
Still, Graham faulted President Barack Obama for failing to signal a commitment to staying in the country as long as it takes.
"At the end of the day the president has to let the Afghan people - the regional players know, the America people know - that we are not going to leave until we are successful," he said. "The president needs to speak more openly about why we can't lose in Afghanistan, to get the American people behind staying and being successful."
Graham had previously argued that Obama's plan for a gradual drawdown would invite insurgents and terrorists to wait until the withdrawal to go on the offensive. But on Sunday Graham acknowledged that a surge of U.S. forces in parts of Afghanistan has begun to create a secure environment for the drawdown.
"If we had no troops it would go back to civil war. But this additional military capacity is beginning to show some effects in certain parts of Afghanistan," he said. "I do see a path forward next summer to transition certain areas of Afghanistan."
Video from Afghanistan: the Road Ahead:
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Petraeus: Iraq "Much More Hopeful Place"
Schieffer also asked Graham about Afghan President Harmid Karzai. Schieffer said that a senator on the Armed Services Committee told him this week that Karzai is a man who still has "a tribal outlook, he has no talent for governing, and he has no strategic outlook."
Graham said that it was time to put Karzai to the test.
"Well, he's an elected leader of Afghanistan. We've recognized the results. Some people are saying they're stuck with Lindsey Graham. That's the way democracy is," Graham said.
"We need corruption trials going forward. We need to show that we're at war with the insurgency. And he needs to rally his people. I think he's capable of doing that. But I'm going to make sure, from a Congress's point of view that we have benchmarks and measurements. It's now time to put him to the test. It's now time to put ourselves to the test because we're running out of time here at home."
Earlier this week CBS Evening News anchor Katie Couric talked to Gen, David Petraeus, the U.S. commander in Afghanistan, about President Karzai. Schieffer noted that the general goes to great lengths not to criticize Hamid Karzai.
Asked whether Karzai has gotten an unfair rap, Patraeus replied: "I think he's in a very, very difficult position. Again, this is Afghanistan and this is a country that is not a developed country. It has a checkered past when it comes to issues such as corruption. I don't think that anyone has alleged that he is party to any of that."