Clint Dempsey’s first goal for Fulham took the Londoners to the brink of Premier League survival as they earned a slender victory over much-changed Liverpool at Craven Cottage.

The American midfielder, signed from New England Revolution in January for £1.5m, chose the perfect moment to open his Fulham account swapping passes with Liam Rosenior down the right, breaking into the box and steering a cool finish past Pepe Reina to secure a priceless three points. It was a backs to the wall effort for Lawrie Sanchez’s side, who were on the defensive for most of the match even though Rafa Benitez made nine changes from Tuesday’s Champions’ League semi-final.

The Liverpool boss fumed afterwards that referee Steve Bennett had missed a headbutt by Michael Brown on Xabi Alonso – with the television replays making appalling viewing – but the visitors squandered a host of chances to put the strugglers away. Craig Bellamy fluffed two early chances to beat Antti Niemi and Robbie Fowler somehow missed an open goal from three yards out after fine play from Jermaine Pennant. Fulham’s openings were few and far between, although Brian McBride miscontrolled when brilliantly found by Phillipe Christanval and Vincenzo Montella’s audacious overhead kick was deflected onto the woodwork.

Fulham had more aggression about them in the second period but struggled to create clear cut chances with Papa Bouba Diop and McBride both nearly scrambling the ball past Reina in one frantic period of play, but Niemi bailed Fulham out at the other end as he dived at Bellamy’s feet to thwart the Welsh forward from close range. The decisive moment came twenty minutes from time. Dempsey showed great invention and desire to work a one-two with Rosenior and drive into the area before keeping his cool as Reina advanced to try and narrow the angle.

There was a scare for the hosts when Gabriel Paletta had the ball in the net with seven minutes to go but the offside flag was quickly raised. Fulham had to survive stoppage time without Diop, who was shown a second yellow card, but they hung on for a massive victory.

FULHAM (4-4-2): Niemi; Rosenior, Bocanegra, Christanval, Knight; Diop, Brown, Davies, Radzinski (Volz 90); McBride (Helguson 86), Dempsey (53). Subs (not used): Lastuvka, Routledge.

BOOKED: Diop, Brown.

SENT OFF: Diop.

GOAL: Dempsey (69).

LIVERPOOL (4-4-2): Reina; Arbeloa, Paletta, Hyypia, Insua (Finnan 75); Pennant (El Zhar 65), Alonso, Sissoko, Gonzalez (Kewell 77); Fowler, Bellamy. Subs (not used): Padelli, Hobbs.

BOOKED: Fowler, Arbeloa.

REFEREE: Steve Bennett (Kent).

ATTENDANCE: 24,554.