According to the report, the killings took place on August 30 in the town of Kahor in the central Afghan province of Daykundi.
Out of the 13 people who were killed, 11 of the victims were former members of the Afghan security forces.
The group had negotiated a surrender and agreed to turn over government military equipment.
Many of the group — who were also fleeing with family members — had left the area, but Taliban fighters caught up with one vehicle and "opened fire on the crowd."
A 17-year-old girl was among those killed in the crossfire that ensued.
The remaining former Afghan security forces then surrendered to the Taliban, who "promptly took them to a nearby river basin and executed them," Amnesty said in the report.
The rights group said it had verified videos and photographs of the dead bodies, as well as speaking with eye witnesses.
Taliban committing 'horrific abuses'
According to Amnesty, the local Taliban-appointed chief of police for the province denied the killings.
Taliban leaders have not yet responded to media requests for comment on the report.
The killings of surrendering forces "appear to be war crimes," Amnesty said in its report.
"These cold-blooded executions are further proof that the Taliban are committing the same horrific abuses they were notorious for during their previous rule of Afghanistan," said Agnes Callamard, Amnesty International's Secretary General.
Taliban threaten female journalists
Concerns for ethnic minorities
Members of the Hazara ethnicity, who are Shiite Muslims, have been frequently targeted by attacks in the past in Sunni-majority Afghanistan.
Hazaras account for around 9% of Afghanistan's 36 million population — making them the country's third-largest ethnic group. They were also targeted by the Taliban during the group's rule over Afghanistan in the 1990s. The Taliban's members are predominantly Pashtun, one of Afghanistan's two largest ethnic groups.
Following the Taliban's takeover of Kabul on August 15, and subsequent declaration of control over Afghanistan, leaders of the hard-line Islamist group made initial promises to offer amnesty to former government forces and employees — as well as to be inclusive towards ethnic minorities and women.
Reports of killings by Taliban fighters following the takeover, however, continue to surface — with journalists, ex-military forces and others facing persecution by the group.