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However, a Will Evans brace put the third-tier side in command before Michael Baidoo and Ronan Curtis scored late on to add some gloss to the scoreline.
Kyle Storer’s side can look back on a memorable run to the first round as 538 supporters followed the club down to Devon.
Plymouth, managed by former Manchester United and England midfielder Tom Cleverley, went close early on as Redman Evans held Aribim Pepple’s low drive from 20-yards.
And Evans then headed wide when left free in the box 10-yards from goal although Evans had his angles covered.
The home side remained on top as Xavier Amaechi curled a free-kick narrowly over the crossbar from 25-yards out before the same player poked a shot wide later in the first half.
However, Plymouth eventually broke the deadlock five minutes after half time when Evans dispossessed Jessy Bavanganga in the box and curled a shot into the bottom left corner.
Evans then made a stunning save to deny Pepple when one-on-one after Evans intercepted George Forsyth’s stray pass.
Argyle doubled their lead on 69 minutes as Evans bundled the ball home from point-blank range after Curtis played a low cross into the box from the left.
And the home side scored a third goal on 78 minutes when Baidoo fired Jared Rendle’s low ball in from the left into the net from 10-yards.
Plymouth struck again just two minutes later as as Curtis lifted a shot over Evans from Harvey White’s through ball.
Alvechurch are next in action against Real Bedford on Saturday, November 14 with kick-off at 3pm at Lye Meadow.
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