Dozens more were seriously wounded at the facility, a key lifeline that has been running "beyond capacity" during fighting that saw the Taliban seize control of the northern provincial capital for several days.
The strike early today left the building engulfed in flames, with photos posted by Doctors Without Borders showing their staff shocked and dazed.
"At 2:10 am local time... The MSF trauma centre in Kunduz was hit several times during sustained bombing and was very badly damaged," the organisation, known by its French initials, said.
The charity said the bombing continued for more than 30 minutes after American and Afghan military officials were first alerted they were being hit.
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"All parties to the conflict, including in Kabul and Washington, were clearly informed of the precise location (GPS coordinates) of the MSF facilities," the statement added.
NATO conceded US forces may have been behind the strike but has not so far commented on the specific claims of MSF, which has long treated the war-wounded from all sides of the conflict.
The incident has renewed concerns about the use of US air strikes in Afghanistan, a deeply contentious issue in the 14-year campaign against Taliban insurgents.
UN rights chief Zeid Ra'ad Al Hussein called for a full and transparent probe, noting that, "if established as deliberate in a court of law, an air strike on a hospital may amount to a war crime."
"This event is utterly tragic, inexcusable and possibly even criminal," he said.
US Defence Secretary Ashton Carter said a "full investigation" is being carried out.
The Afghan defence ministry expressed sadness but in a statement said "a group of terrorists armed with light and heavy weapon... Were using the hospital building as a position to target Afghan forces and civilians".