The result meant the Blues cannot finish in the top four of the Premier League now, meaning they will have to beat Bayern Munich in the Champions League final on May 19 to qualify for the tournament next year.
The home side, who lost 2-1 at Wembley on Saturday, netted three times in nine minutes through a Michael Essien own goal, Jordan Henderson and Daniel Agger to take total control before Stewart Downing missed a penalty on the stroke of half-time.
Chelsea were shocking, in particular John Terry, who was nutmegged three times in the first half, twice by the excellent Suarez.
Downing had also earlier struck the crossbar while the nearest a much-changed Chelsea came to scoring was when Fernando Torres, on his first return to Anfield following his £50million move south last January, also hit the bar from a tight angle.
Chelsea rolled their sleeves up for the start of the second half, with Ramires bundling in Florent Malouda's delivery past a hapless Pepe Reina shortly after the re-start.
The Blues were the architects of their downfall once more though as Ross Turnbull's poor clearance was capitalised on by Jonjo Shelvey, who scored his first Premier League goal with a stunning finish.