Daggers boss John Still enjoyed a triumphant return to London Road by dumping out League One big spenders Posh - a team he managed for 18 months in the mid 1990s.
Nwokeji's 65th-minute header from a cross by substitute Ben Strevens was enough to settle this southern section second-round tie.
Hosts Posh had a Micah Hyde goal disallowed for offside early in the second half and then were guilty of squandering golden opportunities after falling behind.
Wales defender Craig Morgan steered a free header wide from just six yards before former Dagenham striker Craig Mackail-Smith blazed wastefully wide with only goalkeeper Tony Roberts to beat.
Shane Huke and Strevens both went close to doubling Dagenham's lead, but one goal proved enough.