Johnson was signed from third-tier Wolverhampton Wanderers on Monday to ease West Ham's defensive injury crisis that has deprived Allardyce of all his center backs. He couldn't have endured a more miserable debut.
The misery was far from over for West Ham, either.
Negredo completed his second hat trick for City with a snapshot in the 49th after David Silva's low cross deflected into his path and strike partner Dzeko got in on the act on the hour mark, poking home a near-post finish from Gael Clichy's center.
As Allardyce looked on gloomily from the technical area as the rain poured down, Dzeko added the sixth in the 89th with a vicious left-footed shot into the top corner. Allardyce was given a vote of confidence from West Ham's co-chairmen after the Forest defeat but he looked a beaten man here.
"However hard they tried, and they did, was not good enough to stop the quality and talents Man City have bought on the finance that is available to them," Allardyce said.
City has won all 12 of its home games in domestic competition this season and scored 92 goals in 31 games in all competitions.
"We have not arrived in the final but we are very near," Pellegrini said. "It is very important to continue in all the competitions playing the way we did today."