The Taliban, who strongly opposed the US agreement, claimed responsibility for the two early-morning attacks on vehicles carrying military employees to work in the capital.
"There have been two suicide attacks targeting buses carrying Afghan national army personnel," Farid Afzali, chief of the city's police investigation department, told AFP.
The Taliban claimed at least 20 soldiers were killed, but the insurgents regularly exaggerate death tolls after attacks.
"This is a clear message to the stooge government that signed the slave pact, and we will step up our attacks after this," Taliban spokesman Zabihullah Mujahid told AFP.
Afghanistan and the United States on Tuesday signed the long-delayed bilateral security agreement to allow about 10,000 US troops to stay in the country next year.
The signing took place on newly-inaugurated President Ashraf Ghani's first day in office and represented a major step towards mending frayed ties with Washington.
Hamid Karzai, who stood down as president on Monday, had refused to sign the deal - a disagreement that symbolised the decline in Afghan-US relations over recent years.
US-led NATO combat operations in Afghanistan will finish at the end of this year, and the Taliban have launched a series of recent offensives that have severely tested Afghan soldiers and police.
NATO's follow-up mission, which will take over on January 1, will be made up of 9,800 US troops and about 3,000 soldiers from Germany, Italy and other member nations.
The new mission - named Resolute Support - will focus on supporting Afghan forces as they take on the Taliban, in parallel with US counter-terrorism operations.
President Barack Obama welcomed the deal, saying it was an historic day.
He said it provided "the necessary legal framework to carry out two critical missions after 2014: targeting the remnants of Al-Qaeda and training, advising, and assisting Afghan National Security Forces."