Grimsby boss David Artell made six changes for the game and they had to work hard to find the opener through a solid Weston defence which included a lecturer, painter and decorator and a personal trainer among their part-time player ranks.
Jamie Walker’s effort trickled wide of the post, Tyrell Warren’s strike was palmed wide and Vernam sent the ball soaring over the bar among 10 first-half shots for the hosts.
The Seagulls – currently second in their division – had their own efforts, all through the in-form Coulson, who forced Jackson Smith into three saves.
Yet Grimsby found their opener four minutes before the break, after Jude Soonsup-Bell headed on a cross from Vernam, which Harris tipped over the bar to give them a corner.
Vernam took it short and after a one-two unleashed a strike from distance which Harris could this time only watch fly by him into the net.
Weston came out for the second half with impetus as Britton – their goalscoring hero of the second round – shot narrowly over.