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 for the visitors with a tremendous effort before a Mark Noble penalty and Steven Gerrard own goal turned the game in West Ham's favour before the break.
It was another West Ham academy graduate in substitute Joe Cole who pulled the visitors level in the second half with a James Collins own goal completing the comeback for Liverpool.
Gerrard played the ball out to Johnson on the touchline and the England full-back cut inside Jarvisbefore arrowing a strike past Jussi Jaaskelainen for the opener.
West Ham equalised when a set-piece was cleared as far as Guy Demel whose shot hit the arm of Joe Allen. Referee Lee Probert deemed the block to be intentional and pointed to the spot, with Noble stepping up to coolly convert.
A spell of pressure yielded a second goal for Allardyce's side when Jarvis produced another fantastic delivery into the box and the ball was sent flying past Reina off the head of Gerrard.
Cole had been introduced by Liverpool before the game was 30 minutes old and was warmly received by the home fans. The former Chelsea had little impact until he struck a venomous equaliser after being played in by Sterling.
The turnaround was completed just 12 minutes from time as Collins competed for a cross with the balling looping up off the boot of the West Ham defender and over a helpless Jaaskelainen.