The bus was carrying employees of the attorney general's office to work in downtown Kabul when the attacker detonated his suicide vest, as the militants press on with their annual spring offensive.
The Afghan interior ministry "strongly condemned" the attack, confirming that one civilian was killed and 15 others were wounded.
"These attacks... Demonstrate [an] extreme level of atrocity by terrorists against innocent and defenseless civilians," the ministry said in a statement.
The attack comes after a 20-member Afghan delegation yesterday launched two-days of "open discussion" with Taliban representatives in the Gulf emirate of Qatar - in their latest effort to end Afghanistan's long war.
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The Taliban claimed responsibility for today's attack on their Twitter account, claiming that 40 people were killed or wounded.
The Taliban are known to exaggerate and distort their public statements as part of a propaganda drive accompanying their campaign against the Afghan and US-led foreign forces who ousted them from power in 2001.
This year's Taliban offensive marks the first fighting season in which Afghan forces will battle the insurgents without the full support of US-led foreign combat troops.
NATO's combat mission formally ended in December but a small follow-up foreign force has stayed on to train and support local security personnel.
The Taliban launched their spring offensive across Afghanistan late last month, stepping up attacks on government and foreign targets and inflicting a heavy toll on civilians and Afghan security forces.