John Terry and Steven Gerrard scored from set-pieces at either end of the first half - not good enough for the reds to keep up the pressure for 4th place.
The reds, unchanged from last week's nervy win over QPR, gave new Champions Chelsea a guard of honour at Stamford Bridge ahead of the game - and didn't appear to react well when Chelsea powered ahead within the first five minutes.
The reds failed to deal with a corner that Terry fired past Mignolet and Glen Johnson on the line.
Chelsea dominated for large spells - Eden Hazard seeing plenty of the ball as Emre Can (now apparently a right back in a back 4) struggled to contain one of the Premier League's best players.
Liverpool did improve, very slowly, over the course of the half. As half time approached the reds grabbed an unlikely goal. Jordan Henderson took a free kick from just outside the box and at the back post Steven Gerrard had all the space and time he needed to nod a simple header past Courtouis. There was no wild celebration as the Captain ran back for the restart.
The second half saw Liverpool introduce Jerome Sinclair and Jordon Ibe, replacing birthday boy Adam Lallana and Rickie Lambert but it failed to add the extra spark for a dramatic winner.
Steven Gerrard was subbed off with just over 10 minutes left - a mixture of predictable boos and offensive gestures, combined with some applause from the Chelsea supporters. Jose Mourinho amongst those offering applause.
The reds were too sterile for large portions of this game and looked like a team who's season finished some weeks ago - which to be true it had.
Two games left for Liverpool - effectively friendlies now - with Palace at home set to be Gerrard's swansong before an awkward end game against Stoke away.
Some of the crowd are on the pitch - they think it's all over.
It is now.