Hague said Cameron knew about last week's operation, which ended in an embarrassing failure when British personnel were held captive by Libyan opposition groups.
As the secret mission was condemned in the Commons as "ill-conceived, poorly planned and embarrassingly executed", a Whitehall blame-game broke out over the operation, The Telegraph reported.
Guarded by Special Forces troops, British intelligence officers last week arrived near Benghazi by helicopter as a "pathfinder" exercise to prepare the ground for a larger diplomatic delegation.
The secret mission failed when local Libyan forces put the British personnel in "temporary detention". Drawing laughter from MPs, Hague said that was caused by a "serious misunderstanding" about their mission.
The prime minister has faced persistent criticism over his handling of the Libyan crisis, and Labour said the botched "diplomatic mission" has raised fresh questions about the Government's competence, the report said.
Yesterday morning, Downing Street said that Hague was responsible for the mission in the Benghazi area. Its version of events caused anger in the Foreign and Commonwealth office, where diplomats saw it as an attempt to shift blame.
Hague later told MPs that Cameron and other ministers were privy to the secret operation. "The Prime Minister and other colleagues were aware that we would attempt to put a diplomatic team into eastern Libya," Hague said.
The foreign secretary said he took final responsibility for the mission but insisted that the "timing and details" were down to military commanders and Whitehall officials.
Some critics have questioned the decision for the officials to travel by helicopter in the early hours of Friday morning, instead of making the short journey from central Benghazi in the open.
"The timing and details of that are operational matters decided by the professionals, but ministers must have confidence in their judgments as I do and must take full ministerial responsibility for their judgments and decisions, as I do," Hague was quoted as saying by The Telegraph.
Sir Menzies Campbell, the former Liberal Democrat leader, told Hague the failed mission had damaged Britain's international standing.
"I regret what I am about to say. Isn't it clear that this mission was ill-conceived, poorly planned and embarrassingly executed. "What is he going to do to restore the reputation of the United Kingdom in relation to foreign policy in the Middle East?" he said.