Uruguay international Suarez was serving a one-match ban but the Reds struck twice late in the game to seal a win that sees them leapfrog West Ham in the table.
Former Hammers' defender Glen Johnson opened the scoring with a tremendous effort before a Mark Noble penalty and a Steven Gerrard own goal turned the game in West Ham's favour. It was another West Ham academy graduate in substitute Joe Cole who pulled the visitors level with a James Collins own goal completing the comeback for Liverpool.
West Ham's Matt Jarvis was getting a lot of the ball down the left flank but was soon found wanting when he was tested defensively. Gerrard played the ball out to Johnson on the touchline and the England full-back cut inside Jarvis with ease before arrowing a strike past the out-stretched hand of Jussi Jaaskelainen.
Nine minutes before the interval, a West Ham set-piece was cleared as far as Guy Demel whose shot hit the arm of Joe Allen as the former Swansea man closed him down. Referee Lee Probert deemed the block to be intentional and pointed to the spot, with Noble stepping up to coolly convert the penalty.
The spell of pressure yielded a second goal for Sam Allardyce's side, albeit it in fortunate circumstances.
Liverpool switched off after conceding a free-kick which Noble took quickly and found Jarvis in space on the left. The 26-year-old produced yet another fantastic delivery into the box and the ball was sent flying past Jose Reina off the head of Gerrard.
Joe Cole had little impact on the game until he struck a venomous equaliser after being played in by Raheem Sterling and, like Johnson, he refused to celebrate the goal out of respect for his former team.
The turnaround was completed just 12 minutes from time as Jordan Henderson swung in a low cross that Jonjo Shelvey and Collins competed for with the balling looping up off the boot of the West Ham defender and over a helpless Jaaskelainen.