Mujahid, the Taliban spokesman, also claimed responsibility for the Kabul bombing.
The attack occurred three days after a would-be car bomber was shot dead by police in downtown Kabul. That assailant was driving a vehicle packed with explosives and officials said he appeared to be targeting an intelligence agency office.
It also comes as the U.S.-led military coalition in the country is backing off from its claim that Taliban attacks dropped in 2012, tacitly acknowledging a hole in its widely repeated argument that violence is easing and that the insurgency is in steep decline.
Some 100,000 international troops are helping secure Afghanistan at the moment, but most, including many of the 66,000 Americans, are expected to finish their withdrawal by the end of 2014.
Also on Wednesday, the top U.S. commander in Afghanistan, met with Afghan President Hamid Karzai to discuss abuse allegations against American special forces and Afghan troops linked to them in the strategic eastern Wardak province.
The allegations led Karzai to issue an order on Sunday calling for U.S. special forces to be expelled from the province within two weeks despite fears that the move would leave the restive area and the neighboring Afghan capital more vulnerable to al-Qaida and other insurgents.
Karzai and Gen. Joseph Dunford, commander of all U.S. and allied forces, discussed the issue and agreed to work together to address the security concerns of the people of Wardak, a coalition statement said.
Wednesday's meeting came a day after hundreds of Wardak residents converged on the provincial capital of Maidan Shahr to call on Karzai to implement his decision as soon as possible.
"The people are really angry about the actions of both the U.S. special forces and the Afghans working with them," provincial government spokesman Attaullah Khogyani said, adding the protesters threatened to stage larger demonstrations if the elite troops aren't gone by the deadline.