"When the Security Council failed to heed Mr. Annan's repeated calls for collective and significant consequences for non-compliance with its prior resolutions, those members who blocked this action effectively made Mr. Annan's mission impossible," she said in a written statement.
Catherine Ashton, the European Union's foreign policy chief, said Annan's resignation "makes clear how unfortunate it has been that the U.N. Security Council was unable to agree to a resolution."
"The EU continues to support the efforts of the U.N. and the League of Arab States and calls for the early appointment of a successor to carry on Mr. Annan's work towards a peaceful political transition in Syria," Ashton said in a written statement.
Ban, who was first to announce the move, praised Annan's "determined and courageous efforts" and said the news left him with "deep regret." He said that Annan worked within the mandate provided to him by the U.N. General Assembly and that he is "indebted to him and his team for all they have tried to achieve."
But Ban also said the government and the opposition have been determined "to rely on ever-increasing violence," and there have been "persistent divisions within the (U.N.) Security Council."
"Kofi Annan deserves our profound admiration for the selfless way in which he has put his formidable skills and prestige to this most difficult and potentially thankless of assignments," Ban said.
Annan had been secretary-general from 1996 to 2006 and was involved in other initiatives, including the mediation of the election dispute a few years ago in Kenya.
Ban said peace can come only if there's a "firm commitment to dialogue" among the players in Syria and international unity.
"Tragically, the spiral of violence in Syria is continuing. The hand extended to turn away from violence in favor of dialogue and diplomacy -- as spelled out in the six-point plan -- has not been taken, even though it still remains the best hope for the people of Syria."