It comes as the UK pledged ?52.25m in emergency aid to help millions of people affected by the disaster.
Mr Mitchell, who is visiting Kenya, said the money would be used there, as well as in Somalia and Ethiopia.
The World Food Programme estimates 10 million people are affected by the worst drought in over half a century.
And the United Nation's Children's Fund believes about two million young people are malnourished.
The international development secretary said the situation was "getting worse" and urged the international community to do more.
Continue reading the main story Martin Plaut Africa analyst, BBC NewsProviding aid inside Somalia is now a major priority. The UN's refugee agency estimates that nearly one-and-a-half million Somalis have been forced from their homes and are internally displaced by the drought.
Until very recently, helping them had been very difficult indeed. The Islamic militants of al-Shabaab, who control most of the south and the centre of the country, have only recently lifted their ban on outside aid agencies operating in their areas.
Children's fund Unicef has airlifted food to Baidoa, north-west of the capital, for the first time in years. The Red Cross is opening 10 new feeding centres in the Afgoye corridor, near the capital. And medical charity MSF is providing nutrition to children at a spontaneously formed camp of about 5,000 people at Jilib, on the Juba river.
It shows that aid can now get through to areas controlled by Islamists that were, until very recently, out of bounds to the aid agencies.
Mr Mitchell is currently visiting the Dadaab camp in Kenya, which is overflowing with tens of thousands of refugees fleeing the parched landscape in the region where Somalia, Ethiopia and Kenya meet.He estimates that there are about 400,000 people in the camp.
He said: "We need everyone who can help from across the world now to make sure they focus on this developing crisis here to stop it becoming a catastrophe. There is an emergency developing of profound proportions.
"Britain, as always, has shown huge generosity and is in a leadership position to try and resolve this crisis. We need others to do so too. We need the whole of the international community now to bend every sinew to help these poor people here who are in a desperate condition."
Mr Mitchell said the situation is particularly devastating in Somalia, where families already have to cope with living in one of the most insecure countries in the world.
"More than 3,000 people every day are fleeing over the borders to Ethiopia and Kenya, many of them arriving with starving children," he said.
"The international community must do more to help not only refugees but also those victims of the drought who remain in Somalia."
Mr Mitchell also paid tribute to British charities and the public who had donated, saying they had "put their shoulder to the wheel" to try to help the victims.
These sentiments were echoed by Oxfam, which welcomed the government's additional funds and urged other rich countries to be equally generous.
It said: "There is at least a $700m (?434m) black hole in the aid effort which needs to be filled to save lives and avoid a humanitarian crisis becoming a full blown disaster."
The UK's ?52.25m aid package comes after a joint charity appeal by the Disasters Emergency Committee (DEC) saw more than ?13m raised in a week.
The cash is in addition to the ?38m food aid package announced on 3 July to feed 1.3m people for three months.
The Department for International Development (DFID) said the money would help:
500,000 people in Somalia, including treatment for nearly 70,000 acutely malnourished childrenMore than 130,000 people in the Dadaab camps to help provide them with clean drinking water and health care100,000 people in Dolo Ado refugee camps in Ethiopia to provide them with shelter and clean drinking water as well as targeted treatment of starving children300,000 Kenyans, including special rations to prevent malnutrition in children under the age of five and breastfeeding mothers.A spokesman for the United Nations refugee agency, the UNHCR, Ron Redmond told the BBC the Dadaab camp is holding four times the number of people it was designed for and is unable to cope with the volume of people arriving each day.
"We have so many people arriving, in fact about 1,500 a day, that we now have 60,000 people living on the outskirts of the camp... because there is simply no more room inside."
The money will help drought-hit areas of Somalia, Ethiopia and Kenya During his visit to Kenya, Mr Mitchell is meeting the head of the DEC, Brendan Gormley, and Justin Forsyth, chief executive of Save the Children.
Mr Gormley said the need to "scale up" the response to the disaster was urgent, adding that he was pleased that the government had announced more funding.
He said: "Combined with the extraordinary generosity of the UK public to the DEC East Africa Crisis Appeal, we can truly say that the UK is playing a leading role in responding to this disaster.
"There is still, however, a great deal more to be done before we can say we have safeguarded the lives of the 10 million people at risk."
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