A fifth-minute own goal by defender Martin Skrtel kept Fulham ninth in the table, on the same points as Liverpool but with an inferior goal difference.
In the night's other game, seventh-placed Everton drew 1-1 at Stoke City.
Everton went in front with a Peter Crouch own goal on the stroke of halftime but Stoke, who are 13th, equalised through Cameron Jerome midway through the second half.
Chelsea warm up for Saturday's final at Wembley by taking on Newcastle United in the league tomorrow.