The Gunners had slumped to successive league defeats against Manchester City and Chelsea to leave them facing an uphill battle to regain a place in the top four.
Wenger knew there was no more margin for error, but must have feared another night of frustration after falling behind in Wednesday night's rearranged Boxing Day fixture to a fine strike from Jack Collison. However, Lukas Podolski crashed in a 25-yard effort to equalise before the break.
Olivier Giroud then scored his first goal in six games to put the hosts in front just after the restart before German forward Podolski tore the Irons apart, charging down the left to set up Santi Cazorla, Theo Walcott and Giroud again as Arsenal romped out of sight before the hour.
West Ham had to play the closing stages with 10 men after what looked a serious injury to substitute Daniel Potts, who was carried off on a stretcher following a serious head injury and kept in hospital overnight for observation.
"I never felt that the players didn't know how important it is. I was more worried that they are sometimes a bit inhibited in the recent big games from the start on," Wenger said.
"But this team has a fantastic mentality and good quality, that's what we want to show until the end of the season now."