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Kenny Jackett's men rarely played like a team challenging at the top and even the inclusion of their new signings failed to spark the side to victory.
Grimsby had a shout for a spot-kick in the sixth minute when Michael Reddy went down in the box after a challenge, but referee Scott Mathison waved played on.
Two minutes later Jason Crowe worked his way through but home keeper Willie Gueret was at full stretch to push the ball behind for the first corner of the game.
It was not until midway through the first half that the Swans forged a clear opening, but Paul Connor's shot on the turn was safely gathered by Anthony Williams in the Mariners' goal.
Five minutes before the interval Martin Gritton caused problems in the home defence and it needed a clearance off the line by Andrew Gurney to prevent the visitors going ahead.
There was relief all around when the referee blew the half-time whistle as it had been a dire opening period littered with errors from both sides.
The second half began in the same fashion with Grimsby looking far more dangerous on the break with Reddy and Gritton lively front runners.
A mistake by substitute Alan Tate on the hour gave Reddy yet another chance but he failed to even get a shot on goal.
Lee Thorpe and Kevin McLeod were brought on by Jackett in an attempt to grab a victory, but the performance hardly deserved it.
Former QPR man McLeod made a chance for Thorpe in the 71st minute when he curled in a dangerous cross but he glanced a point-blank header wide of the mark.
Swansea forced a few corners in the closing stages but the goal they so desperately needed never came and they had to be content with a point.