Albion lead through a sixth-minute Deniz Undav goal.
Albion have dominated a slow-paced match with Kaoru Mitoma and Pascal Gross spurning good chances to increase Albion’s lead.
Mitoma scooped an effort over from almost under the cross bar.
Moises Caicedo also saw an effort fly over the bar.
Grimsby’s best chance came from an over-hit Lewis Dunk back pass which saw Robert Sanchez scurrying back towards his goal to clear.
If the scoreline holds, the Seagulls are just 45 minutes from a Wembley semi-final.
