The death on Saturday was the 10th US service member to die under hostile fire and the 12th overall this year.
Mujahed said two US service personnel were killed when Taliban militants attacked a tank in Sayed Abad district of central Wardak province, barely 40 miles south of the Afghan capital, Kabul. "All sides also welcomed intra-Afghan meetings held in Moscow and Doha", the statement said.
More than 2,400 American military personnel have been killed since the war began and more than 20,000 others have been wounded.
Between January and March this year, air operations by Afghan and global forces have caused at least 145 deaths, accounting for almost 25 percent of the total deaths during that period, UN Assistance Mission in Afghanistan (UNAMA) said in its quarterly report in April.
Separately, Afghanistan's Islamic State affiliate claimed Friday's suicide attack that killed 6 people.
USA military statement announcing the killings of the US service members was a terse, two-paragraph announcement.
The NATO-led Resolute Support said it would not release the name of the soldier or give further details until 24 hours after giving notification of the death to the soldier's family.
They said they had blown up an "American tank" in the Sayed Abad district of Wardak province.
Pompeo added that discussions with the Taliban have also begun on USA troop withdrawal.
Pakistan joined the three powers in talks in Beijing that come as the United States moves closer to an agreement with the Taliban to pull troops from Afghanistan and end its longest-ever war.
"While we've made clear to the Taliban that we are prepared to remove our forces, I want to be clear we've not yet agreed on a timeline to do so", Pompeo said.