Liverpool hit the woodwork four times but looked to be set for a second goalless draw this season against the Londoners until Benayoun came off the bench to lash in the winner.
Chelsea [ Images ] are third on 64 after goals by Frank Lampard [ Images ] and Florent Malouda [ Images ] early in the second half beat Newcastle United 2-0 to spoil the St James' Park comeback of temporary manager Alan Shearer and leave his club in real relegation danger.
Arsenal's [ Images ] long-term absentees Cesc Fabregas [ Images ] and Emmanuel Adebayor combined to create two goals for the Togo striker in a comfortable 2-0 home win over Manchester City [ Images ] that consolidated their fourth place on 58 points.
Fulham, who had beaten Arsenal and Manchester United at Craven Cottage this season, were outclassed on Saturday but somehow stayed in the game.
Liverpool had put 13 goals past Real Madrid [ Images ], United and Villa in their last three games but Fulham rode their luck.
Andrea Dossena glanced a header against the bar, Xabi Alonso [ Images ] rattled it from the edge of the box, Fernando Torres [ Images ] hit a post and, when Dossena sent another header against the bar, Liverpool must have begun to wonder if it would be their day.
Fulham tightened up in the second half and, after surviving a few more scares, seemed set for a point before Liverpool struck in stoppage time for the eighth time this season.
"It's a massive three points, we knew a draw was no good to us. We had to win to put some pressure on United," Gerrard told Setanta Sports. "We've got to let them know they are in for a title fight all the way to the end.
"There were times when you think it's not going to be our day but Yossi pops up with important goals and that goal could be big for us."
Chelsea were delighted to have negotiated what looked an awkward hurdle as the return of Shearer seemed certain to galvanise a side that had won only once in 12 league games.
Newcastle held their own in the first half but a defensive blunder by Argentina centre back Fabricio Coloccini let in Nicolas Anelka [ Images ] and after the Frenchman's shot came back off the bar, Lampard was on hand to nod the ball in after 56 minutes.
The second goal was even more basic as Anelka flicked on a goalkick and Lampard set up Malouda to shoot home.
"With the coming of Shearer we knew it would give a lot of energy and we wanted to spoil the party," Chelsea manager Guus Hiddink [ Images ] told the BBC.
Shearer said: "Effort-wise there's no problem, the response from the players and everyone over the last few days has been great. There is a lot of hard work to be done, we haven't got a lot of time, but I think we'll get there."
Newcastle are now three points from safety following good results for most of the teams immediately above them.
Blackburn Rovers beat Tottenham Hotspur [ Images ] 2-1, Stoke City [ Images ] claimed their first away win of the season with a 2-0 triumph at West Bromwich Albion [ Images ], Bolton Wanderers thrashed Middlesbrough 4-1 while Hull City and Portsmouth drew 0-0.
The foot of the table makes grim reading for the three north-east clubs. West Brom remain bottom on 24 points with Middlesbrough (27), Newcastle (29) and Sunderland, on 32 after losing 2-0 at West Ham United, filling the next three places.