Gen. Mohammad Daud Amin, Kabul's deputy police chief, said intended target Mohammad Mohaqiq, a lawmaker and former cabinet member from the minority Hazara community, survived the attack, but at least four bodyguards were among the nearly two dozen people wounded, the Los Angeles Times reported.
No group claimed immediate responsibility.
Mohaqiq told the Los Angeles Times in a phone interview he was on his way to Parliament when the bomb exploded.
"I think the Afghan security forces are not in a position to provide security nationwide and need more training," he said.
Tuesday's attack came just before the formal transfer of security responsibility to Afghan National Security Forces, a milestone in the 12-year-old war. Afghan President Hamid Karzai and NATO Secretary General Anders Fogh Rasmussen attended.
U.S. Defense Secretary Chuck Hagel issued a statement on the transfer of responsibility praising the sacrifices all parties have made.
"This critical milestone is a tribute to the determination of the people of Afghanistan to take responsibility for their country's future, and it was made possible by heavy sacrifices on the part of the United States, coalition partners, and the Afghan forces," he said. "This achievement keeps us and our coalition partners on track to bring our combat mission to a close next year and transition to a non-combat, train, advise and assist mission that will help ensure Afghans can sustain security into the future."
Tuesday's ceremony was the fifth and last stage of a two-year transition to build up confidence and capability of Afghan forces.
Going forward, even if American units are close by, Afghan forces must operate without U.S. air support, medical evacuation helicopters or partnered combat units, The New York Times said. The U.S. military will provide air support to Afghans only when an exception is approved by an officer holding a general's rank.
The U.S. military has provided equipment and training to help its Afghan counterparts use mortars and other artillery, said Maj. Gen. James C. McConville, the U.S. commander in eastern Afghanistan.
"If you do it for them, they will never build the capability and the capacity to do it," McConville said. "We don't want them to get used to a capability they're not going to have in the future."