"I do not ever comment on intelligence matters but I can say that we are helping elements of the Syrian opposition, but in a practical and non-lethal way," he said.
"We have helped them with communications and matters of that kind, and we will help them more."
Mr Annan resigned as the international peace envoy for Syria on Thursday night, criticising the UN Security Council for "finger-pointing and name-calling" when it should have been taking action to resolve the crisis.
Mr Annan, the former secretary general of the United Nations, placed most of the blame on the Syrian government's "intransigence" for his departure, which effectively closes the main diplomatic avenue for resolving the country's civil war. Mr Annan's resignation will take effect on Aug 31.
Western governments are now expected to reconsider their options, with America and Turkey believed to be discussing whether to allow surface-to-air missiles to be supplied to Syrian rebels as a way of hastening the regime's demise.
Russia and China have blocked British-backed resolutions in the UN security council to threaten the Assad regime with global sanctions if it fails to halt the bloodshed in the 17-month uprising.
Mr Hague said Mr Annan's departure should not, however, halt efforts to secure a UN agreement.
"It is not dead but it is a bleak moment not only for the people of Syria but for our diplomatic efforts. Diplomacy has so far failed the people of Syria," he said.
"We do not give up on the diplomacy with Russia and with China; we will keep going with that as long as this situation continues but we will have to do other things as well," he said.
The 193-member UN General Assembly is expected to approve a resolution today denouncing the use of major military hardware against opposition forces and demanding chemical and biological weapons not be deployed.
But the text tabled by Saudi Arabia has been stripped of calls for the resignation of Mr Assad and for wider sanctions amid pressure from countries such as Russia, which said it was "unbalanced".
