Dominic Raab said those responsible would be held to account and that it was “essential to defend against these deplorable acts”.
At least 12 people were injured in the attack on the Taji military camp, north of Baghdad.
It came amid heightened tensions between the US and Iran.
The Iraqi military has opened an investigation into the attack.
It coincided with what would have been the birthday of Iranian general Qasem Soleimani, who was killed in a US drone strike in January.
Taji air base is used as a training site for coalition forces.
The British soldier who was killed was a member of the Royal Army Medical Corps, the Ministry of Defence said.
US military sources said an American soldier and an American contractor were also killed. No names have yet been released.
Mr Raab offered his “heartfelt condolences to the families of those killed in this cowardly attack”.
He added that he had discussed the attack – and how to respond – with US Secretary of State Mike Pompeo.
Mr Raab said: “We agreed that it is essential to defend against these deplorable acts. We must find those responsible.”
Prime Minister Boris Johnson said: “We will continue to liaise with our international partners to fully understand the details of this abhorrent attack.”
Speaking to BBC News, defence minister Johnny Mercer said the UK’s commitment to peace in Iraq remains despite the “absolute tragedy”.
“I think that we should continue to do everything possible to keep this country safe,” he said.
“Where that requires us to partner with coalition forces in a fight against a deadly enemy like Daesh, I think we should continue to contribute to that mission.”
US military spokesman Colonel Myles Caggins said 12 people from the Combined Joint Task Force Operation Inherent Resolve were injured when more than 15 small rockets hit the base on Wednesday at 19:35 local time (16:35 GMT).
Mr Pompeo said the attack would “not be tolerated” and that the UK and US have agreed that “those responsible must be held accountable”.
About 400 British troops are stationed in Iraq, while the US has 5,200.
The Army said that British troops are in the country to provide training and equipment to Iraqi and Kurdish security forces – rather than in a combat role – and have trained more than 25,000 Iraqi forces.