Two goals by Crystal Palace forward Christian Benteke, who Liverpool sold to their opponents earlier this season, cost the hosts the chance to put pressure on the Premier League's top two with a rare home defeat on Sunday.
Philippe Coutinho had put Liverpool ahead in the 24th minute with a direct free kick.
FA Cup results
Chelsea 4, Tottenham Hotspur 2
Arsenal 2, Manchester City 1
Premier League
Sunday's results
Burnley 0, Manchester United 2
Liverpool 1, Crystal Palace 2
Saturday's results
Swansea 2, Stoke 0
West Ham 0, Everton 0
Bournemouth 4, Middlesbrough 0
Hull 2, Watford 0
The home side dominated but conceded an equaliser just before halftime when Benteke, sold by Juergen Klopp in August, scored from a cross by Yohan Cabaye. In the 74th minute the Belgian stooped to head in a corner and win the game.
Klopp's team stay third but are five points behind second-placed Tottenham Hotspur having played two games more.
Palace, who have recently beaten Chelsea and Arsenal, moved up four places to 12th and should now be safe from relegation after a sixth win in eight games.
At Wembley Stadium, Alexis Sanchez ensured Arsenal's stormy season will end with an all-London FA Cup final against Chelsea, after securing a 2-1 victory over Manchester City in extra time.
Chilean striker Sanchez struck in the 101st minute to secure the all-London May 27 FA Cup final against Chelsea. Arsenal are chasing a third title in four seasons in the competition that has provided a comfort blanket for Arsene Wenger during a 13-year Premier League trophy drought.
But with Arsenal languishing in seventh place in the Premier League, and set to miss out on the Champions League, the heat remains on the manager whose contract expires at the end of the season.
"People can question my decisions but not loyalty and commitment to this club," Wenger said coyly about his managerial future after goals from defender Nacho Monreal and Sanchez gave him a shot at a seventh FA Cup triumph next month.