A Reuters witness saw the body of a middle-aged man lying at the entrance to the building.
"We're deeply distressed by this loss," International Committee of the Red Cross (ICRC) spokesman Ewan Watson told Reuters.
The ICRC offices in Donetsk are in a three-storey building less than one kilometre from the state security headquarters which has been occupied by separatist rebels along with other strategic points in the city since April.
The incident will bring fresh strain to bear on a fragile ceasefire called by Ukrainian President Petro Poroshenko which came into force on Sept. 5.
Poroshenko said last week the ceasefire was holding. But it has come increasingly frayed in the past few days with increased fighting going on around the main international airport of Donetsk, from where separatists are trying to dislodge government forces.
Ukrainian Foreign Minister Pavel Klimkin blamed the separatists for Thursday's attack. "I have only one question: do the terrorists have any idea of what humanity is all about when they shell the Donetsk office of the ICRC, whose only aim is to help people?" he said.
There was a shell crater in the pavement close to the body of the dead man and pools of blood. Windows were shattered and debris from the blast was strewn on the ground.
A van arrived after a while and two men put the body on a stretcher and drove it away.
Seven Ukrainian soldiers were killed in a single strike by tank fire on their armoured personnel transporter on Monday. On Wednesday at least 10 people were killed when shelling hit a school playground in Donetsk and a public transit mini-van in a street nearby.
