The man, wearing a black overcoat, approached the bus purposefully in heavy morning snow as soldiers were boarding, set down his umbrella and went under the chassis as if to fix something, according to a witness.
Watching from across the street, office worker Ahmad Shakib said he thought for a moment the man might have been a mechanic.
"I thought to myself, what is this crazy man doing? And then there was a blast and flames," that engulfed the undercarriage, he said. "It was a very loud explosion. I still cannot really hear."
Kabul police said the attack, which was the second this week, wounded at least six soldiers and four civilians. The bomber also died. Bakery owner Mirza Khan said the blast shattered the windows of his nearby shop where people were waiting to buy bread, leaving six wounded.
The Afghan government uses buses to ferry soldiers, police and office workers into the city center on regular routes for work, and the vehicles have been a common target for insurgents, who were also behind this attack.
Taliban spokesman Zabiullah Mujahid claimed responsibility for the bombing in a text message to The Associated Press.
The attack comes 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.
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