Cheney, who visited with President Hamid Karzai and then left the country two hours after the blast, said the attackers were trying "to find ways to question the authority of the central government."
The vice president had spent the night at the sprawling Bagram Air Base, and it was about 10 a.m. when the explosion sent up a plume of smoke visible by reporters accompanying. U.S. military officials declared a "red alert."
"I heard a loud boom," Cheney told reporters. "The Secret Service came in and told me there had been an attack on the main gate."
He said he was moved "for a brief period of time" to one of the base bomb shelters near his quarters. "As the situation settled down and they had a better sense of what was going on, I went back to my room," Cheney added.
Asked if the Taliban were trying to send a message with the attack, Cheney said that fighters "clearly try to find ways to question the authority of the central government."
"Striking at the Bagram (base) with a suicide bomber, I suppose, is one way to do that," he said. "It shouldn't affect our behavior."
CNN has more here: The Taliban claimed responsibility and said Cheney was the target.
Maj. William Mitchell said it did not appear the explosion was intended as a threat to the vice president.
"He wasn't near the site of the explosion," Mitchell said. "He was safely within the base at the time of the explosion."
However, a purported Taliban spokesman, Qari Yousef Ahmadi, said Cheney was the target of the attack.
Taliban: 'Attacker was trying to reach Cheney' "We knew that Dick Cheney would be staying inside the base," Ahmadi told The Associated Press by telephone. "The attacker was trying to reach Cheney."
Ahmadi said the attack was carried out by an Afghan, Mullah Abdul Rahim, from Logar province.