The Taliban claimed responsibility and said Cheney was the target.
Talking to reporters later, Cheney said he heard "a loud boom" and was informed by Secret Service agents that there had been an explosion.
He was moved briefly to a bomb shelter at the Bagram base, he said, but returned to his room "as the situation settled down."
"I think [the Taliban] clearly try to find ways to question the authority of the central government," he said. "Striking at the Bagram [base] with a suicide bomber, I suppose, is one way to do that ... It shouldn't affect our behavior."
The vice president met with President Hamid Karzai in the capital, Kabul, about two hours after the bombing, before leaving the country.
There were conflicting reports on the death toll.
Karzai's office said 23 people were killed, including 20 Afghan workers at the base. Another 20 people were injured, it said.
A statement from NATO's International Security Assistance Force said initial indications were that three people were killed including a U.S. soldier, an American contractor and a South Korean soldier. U.S. officials indicated that they planned to update that death toll but wouldn't immediately say what the new figure was.
Associated Press reporters at the scene said they saw at least 12 bodies in black body bags and wooden coffins being carried from the base area to the market, where hundreds of Afghans had gathered to mourn.
Friends and relatives cried and moaned as they carried or drove the bodies away from the base. Two men came to the base entrance crying and wringing their hands, one of them screaming, "My brother!"
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."
Mitchell noted that Cheney's overnight stay occurred only after a meeting with Karzai on Monday was canceled because of bad weather.
"I think it's a far-fetched allegation," he said, referring to the Taliban claim. "The vice president wasn't even supposed to be here overnight, so this would have been a surprise to everybody."
The explosion happened near the first of at least three gated checkpoints that vehicles must pass through before gaining access to Bagram, meaning the attacker did not get near Cheney's location.
The sprawling base houses 5,100 U.S. troops and 4,000 other coalition forces and contractors. High security areas within the base are blocked by their own checkpoints. It was unclear how an attacker could expect to penetrate the base, locate the vice president and get close to him without detection.
"We maintain a high-level of security here at all times. Our security measures were in place and the killer never had access to the base," said Lt. Col. James E. Bonner, the base operations commander. "When he realized he would not be able to get onto the base he attacked the local population."
It was not the first attack apparently aimed at a top U.S. official in Afghanistan. In January of last year a militant blew himself up in Uruzgan province during a supposedly secret visit by the U.S. ambassador, killing 10 Afghans.
Blast shook market 500 yards away
Khan Shirin, a private security guard, sobbed near the dead body of his relative, Farvez, a truck driver and the representative of a transport association that hauls goods for the U.S. base. Shirin said many of the people killed were truck drivers waiting to get inside the base.
Ajmall, a shopkeeper, said the "huge" blast shook a small market where he has a stall about 500 meters (yards) from the Bagram base. Ajmall, who goes by one name, said those wounded in the blast were taken inside the U.S. base for treatment.
The explosion sent up a plume of smoke visible to reporters inside the base traveling with Cheney, and American military officials declared a "red alert" inside the base.
Cheney had spent the night at Bagram and ate with soldiers at the base, telling reporters that "breakfast was excellent" but making no other comments. He was accompanied by Maj. Gen. David Rodriguez, the commander of U.S. troops in Afghanistan.
Meeting topic: The surge in violence
He left the base after the 10 a.m. (12:30 a.m. ET) blast and flew to Kabul, 50 kilometers (30 miles) south of Bagram, to meet Karzai. Cheney was met by armed guards with guns drawn on the tarmac and was rushed by ground convoy to the presidential palace, where he and Karzai walked a long receiving line on Oriental rugs laid out on the wet, stone pavement.
Cheney and Karzai were expected to discuss the surge in violence in Afghanistan. Five years after their fundamentalist regime was toppled, Taliban-led militants have stepped up attacks. Afghan, U.S. and NATO forces are bracing for a fresh wave of violence in the spring.
There were 139 suicide bombings last year, a five-fold increase over 2005, and Rodriguez has said he expects the number of suicide bombs to rise even further in 2007.
The United States has 27,000 troops in Afghanistan. About 14,000 are part of the 35,000-member NATO force commanded by U.S. Gen. Dan McNeill.
Cheney traveled to Afghanistan after a stop Monday in Pakistan.
