The British organisations said in a report, titled "A Dangerous Delay", that a "culture of risk aversion" among aid agencies and national governments caused a six-month delay in taking action.
Oxfam chief executive Barbara Stocking said poor families in East Africa, mainly in Somalia, were still "bearing the brunt" of the failure to mount an effective response to the food crisis.
"We all bear responsibility for this dangerous delay that cost lives in East Africa and need to learn the lessons of the late response," Stocking said.
"We know that acting early saves lives but collective risk aversion meant aid agencies were reluctant to spend money until they were certain there was a crisis."
The report quoted British government figures as saying that between 50,000 and 100,000 people died in the East Africa food crisis, more than half of them children under five.
A US government estimate said more than 29,000 children under the age of five died from May to July 2011.
The report by the two aid organisations said early warning systems forecast a crisis in the region as early as August 2010 but a full-scale response was not launched until nearly a year later in July 2011.
But many donors "wanted proof of a humanitarian catastrophe" before acting, it said.
Save the Children chief executive Justin Forsyth said the suffering of thousands of youngsters could have been avoided with "more money when it really mattered".
"We can no longer allow this grotesque situation to continue, where the world knows an emergency is coming but ignores it until confronted with TV pictures of desperately malnourished children," he added.
