Liverpool are now two points clear of United and although they have played two games more, Rafa Benitez will believe his men can end their 19-year wait for a league title.
It looked as if Liverpool would be made to rue a number of missed chances, but with time running out, Benayoun popped up at the first post to lash a stunning effort past the outstretched arm of Mark Schwarzer.
Liverpool should have been at least four goals ahead at the break, but the Fulham goal lived a charm life with the woodwork coming to their rescue on four separate occasions.
Schwarzer had already pulled off a wonder save to deny Andrea Dossena, before the Italian smashed a header against the crossbar after connecting with Martin Skrtel's shot.
The visitors continued to pile on the pressure with Benitez's men hitting the frame of the goal three times in four mad minutes.
First, Xabi Alonso's rasping 30-yard volley smashed against the crossbar with Schwarzer beaten, before Fernando Torres raced clear only to thunder a shot against the base of the post.
Dossena was next to go close, this time directing a thumping header against the bar after Steven Gerrard's inch-perfect cross found the defender all alone in the Fulham penalty area.
Liverpool must have wondered how they had failed to make the breakthrough and they were further frustrated in the second half as Fulham refused to buckle.
Ryan Babel, a second-half substitute, saw his fierce effort hacked off the line by John Paintsil as Liverpool increased the tempo, while Gerrard's cross-cum-shot was deflected just past the post by team-mate Benayoun.
The Reds poured forward in the dying minutes and when Gerrard's pass was deflected into the path of Benayoun, the Israeli showed composure to smash home a deserved winner.