England ace Rashford watches Squires Gate's FA Cup win
Rashford was among the 200-plus spectators as Gate moved into a 3-0 lead against hosts who had their keeper sent off, though West bravely battled on.
But although they hit back with two goals, the second one five minutes into stoppage time came too late to give Gate any late nerves.
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Hide AdBrett Dawson almost gave Gate the lead after just six minutes, heading against a post from Salthouse’s cross.
Gary Pett’s shot trickled wide 10 minutes later from a cross by Ryan Charnley, who broke the deadlock on 22 minutes, firing home the rebound after Dawson’s volley was blocked.
Gate had their first scare when Adam Sumner was called upon to clear a header off the line but the visitors were two up on 34 minutes.
Joe Booth’s accomplished through-ball met the run of Dawson, who fired past the keeper.
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Hide AdDawson had another opportunity six minutes later but headed Sumner’s corner wide.
It looked to be all over early in the second half, when the West keeper was sent off for handling deliberately outside his area and Booth scored from the resulting free-kick on 52 minutes.
The home side refused to surrender and forced Ben Fletcher to save with his feet on the hour.
And the hosts pulled one back with a pinpoint free-kick by Joe Shaw on 74 minutes.
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Hide AdFletcher made another quality save in stoppage time but could not deny Saul Henderson a second for West, though it is Gate who will contest the preliminary round on Saturday week.
Gate: Fletcher, Richards, Rossall, Sumner, Riley, Booth (c), Ing, Charnley, Salthouse, Pett, Dawson; subs: Bamba, Mullen, Gibson.