Curtis struck with 16 minutes to go to deny Mariners' caretaker boss Stuart Watkiss what would have been a memorable first win in charge of the ailing club, who this week sacked Alan Buckley after a strong of poor results.
Watkiss would certainly have visions of collecting the maximum points after his side kept a lifeless Morecambe at bay all too easily in the first half after taking the lead themselves after just nine minutes from the penalty spot.
Peter Till was the goal hero for the visitors, as he kept his cool to beat Barry Roche after Ryan McCann had fouled Danny Boshell.
The game then became a turgid affair which saw both sides struggle to make any headway as the defences remained well on top.
With Grimsby happy to defend Morecambe pressed, but too often tried the long ball with midfielders failing to get up in support of the front two and they were reduced to only one first-half attempt on goal as Stewart Drummond floated a free-kick just over the top.
The second half began in similar fashion as Morecambe dominated the possession, but failed to threaten the Grimsby goal which was well protected.
But the pattern of the game changed dramatically on the hour when Morecambe brought on Curtis and Craig Stanley.
Stanley's introduction immediately sparked a more energetic display and Curtis added a bit of power up front to supplement Rene Howe's tireless endeavours.
It was quite fitting that the pair combined to bring the Shrimps level on 74 minutes as Stanley curled in a dangerous free-kick at pace and Curtis' flick header beat the flat-footed Phil Barnes to his left.
Curtis should have had a second three minutes later when found by Garry Hunter's superb through ball, but he drilled it over the bar with a shot on the run from 12 yards.
It looked like the visitors would restore their lead when Roche misjudged a corner and it was met by the head of Grimsby skipper, Matt Heywood, but his header was blocked on the line by an alert Hunter.
As the game opened up Roche then made a superb one-on-one save from Nick Hegarty, who had raced on to a through ball beyond any covering home midfielders in the final minute.