Wrexham are now unbeaten in six games after this dire stalemate at the Racecourse.
Substitute Michael Proctor had the best chance of the game at the death but fired his effort narrowly over the crossbar.
Chances were few and far between in a game that lacked life on a pitch that was covered in sand.
Wrexham are six points away from safety after the four teams above them all lost.
Wrexham: Ward, Bolland, Evans, Mike Williams, Spender, Whitley (Aiston 63), Sonner, Llewellyn, Tremarco, Broughton (Proctor 66), Nicholson (Duffy 66).
Subs Not Used: Anthony Williams, Hope.
Booked: Duffy.
Grimsby: Barnes, Bennett, Fenton, Atkinson, Clarke, Boshell, Hunt, Toner (Bolland 46), Hegarty, Till (Jones 68), North (Bore 80).
Subs Not Used: Montgomery, Whittle.
Booked: Atkinson, Fenton.
Att: 4,217
Ref: Andy Haines (Tyne & Wear).
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