(CNN) -- Martin Skrtel's early own goal gave Fulham a historic first win at Anfield on Tuesday and left Liverpool in danger of finishing outside the top eight in the English top flight for the first time since 1954.
With Saturday's FA Cup final against Chelsea in mind, Reds manager Kenny Dalglish rested a host of first-choice players, including Luis Suarez, Steven Gerrard and goalkeeper Pepe Reina.
The home side made the worst possible start when Skrtel headed in former Liverpool defender John Arne Riise's cross, and couldn't recover against a Fulham team that had won only three previous away games this season.
The London club moved level on points with eighth-placed Liverpool, while Everton edged closer to finishing above their Merseyside rivals for the first time in seven years with a 1-1 draw at Stoke.
An own goal by Stoke striker Peter Crouch -- the 1,000th goal of this Premier League season -- gave Everton the lead after 44 minutes before fellow forward Cameron Jerome came off the bench to snatch a point in the 69th minute.
Everton lost to Liverpool in the FA Cup semifinals, but the draw left David Moyes' side three points above the Reds with two league games to play.
Liverpool were booed off by their home fans, having won three of the last 13 league games and only five times at home in the Premier League all season.
"Our attitude wasn't right, and if you go in with a bad attitude you're going to get a bad performance," said Dalglish, whose team won the English League Cup in February.
In La Liga on Tuesday, Villarreal secured a vital 3-2 win at Sporting Gijon that moved them closer to safety and all but condemned the opposition to the second division.
Marcos Senna put the visitors 2-1 up from the penalty spot just before half time and Hernan Perez added a third in the 56th minute before a late consolation from Alejandro Galvez.
The end of the game was marred by the home fans throwing objects onto the pitch as they saw their team slump to a loss that leaves them needing to win their last two games and hope Rayo Vallecano lose both theirs.
Villarreal will be safe if Zaragoza do not beat Levante on Wednesday, as will Granada who beat Espanyol 2-1 thanks to a first-half double by Nigerian striker Odion Ighalo.
Getafe could only draw 1-1 at home to relegated Racing Santander to damage their chances of qualifying for the Europa League.
In Serie A, Roma seem destined to miss out on Champions League qualification after playing out a 0-0 draw in torrential rain at Chievo.
If they are to finish third, Roma need to win their last two games and hope neither Napoli, Lazio, Inter Milan nor Udinese take more than three points from their final three matches.