Warren Joyce's side collected a third consecutive clean sheet to maintain a push for the semi-finals of the competition as Nick Powell netted towards the start and end of the match.
The Reds made the perfect start when Larnell Cole fed Marnick Vermijl on the right and the full-back cut inside to shoot with his left foot. When Raphael Siegel could only push the ball out, Powell pounced to tuck home the rebound.
The visitors responded by creating some chances of their own with Paul McCallum causing problems but Dan Potts needed to be alert to head a long kick by Anders Lindegaard back to Spiegel. Cole then had a shot blocked after an incisive move with Jordan Spence clearing before Powell could profit.
Lindegaard watched a dipping attempt by Potts clear the bar and the defender, son of Steve Potts, escaped a with a clear handball inside the box at the other end. As United gained control, the impressive Powell cut a shot just wide after a neat one-two with Jesse Lingard and was unable to get his head to an accurate long ball by Tom Thorpe as half-time approached.
McCallum was rightly cautioned for elbowing Tyler Blackett after the interval as play became a little scrappy with the Hammers well drilled in their approach. Yet the hosts began to dominate as Powell was almost gifted a second goal by Spence but Spiegel was out quickly to deny the lone striker. Sean McGinty's header from a corner was then safely clutched by the Swiss keeper.
West Ham attempted to mount a late rally but failed to really trouble Lindegaard, thanks largely to the dominance of Thorpe and McGinty at the heart of the defence. Potts did hammer one shot