(adds quotes by Renzi, EU Timmermans, IOM)

By Stephanie Nebehay

GENEVA Feb 12 (Reuters) - The European Union must establish a full search-and-rescue operation in the Mediterranean following the death this week of hundreds of migrants who were trying to reach Italy, the U.N. refugee chief said on Thursday.

More than 300 people are feared to have died off the Libyan coast after they set sail in inflatable dinghies in rough, wintry conditions.

The tragedy came just weeks after Italy wound down its major sea rescue mission, dubbed Mare Nostrum, because of funding concerns, and replaced it with a pan-European operation, Triton, which has fewer ships and covers a smaller area.

"There can be no doubt left after this week's events that Europe's Operation Triton is a woefully inadequate replacement for Italy's Mare Nostrum," U.N. High Commissioner for Refugees Antonio Guterres said in a statement.

This week's death toll included 29 people who died of hypothermia after spending 18 hours on the deck of one of the two, small Italian coast guard vessels which picked them up.

Italian Prime Minister Matteo Renzi, speaking to reporters in Brussels before a meeting of EU leaders, said the "top priority" was to resolve the chaotic situation in Libya, the main transit country for migrants.

"We need to solve that issue if we want to avoid that people die and that the Mediterranean Sea becomes a cemetery," he said.

Frans Timmermans, first vice-president of the European Commission, said that the bloc had a joint responsibility for the lives of desperate people seeking a safe haven.

"I'm calling on all member states to work with the Commission to find comprehensive solutions to better manage migration. We are all in this together," he said in a statement.

Guterres, a former prime minister of Portugal, said: "The focus has to be about saving lives. We need a robust search-and-rescue operation in the Central Mediterranean, not only a border patrol."

If Europe does not implement a wider operation "it is inevitable that many more people will die trying to reach safety in Europe," he said.

Migrants and asylum-seekers from Syria, Iraq, and sub-Saharan Africa have been departing from Libya aboard rubber dinghies in severe winter conditions, marking a deadly early start to the migration season, aid agencies say.

Many of the latest victims were from Senegal, Ivory Coast, Gambia and Mali, they said.

"Once channels open, people opportunistically hear a group is going. The numbers create momentum and then it becomes unstoppable," said Joel Millman of the International Organization for Migration (IOM).

(Additional reporting by Steve Scherer in Rome and Francesco Guarascio in Brussels; Editing by Crispian Balmer)