An edgy opening 20 minutes saw Salford spurn the chance to go ahead, as Kelly N’Mai’s well-timed through ball in the Grimsby box was wasted by Ryan Graydon’s tame strike.
Other than that moment, there was little for either set of supporters to get excited about in a tepid first half but the second came alive soon after kick-off.
Salford’s opener came when Grimsby failed to pick Udoh up at a corner as he latched onto a loose ball and drove home his volley from a tight angle.
The hosts then should have wrapped the tie up minutes later when Ade Oluwo’s header from Luke Garbutt’s corner cannoned off the post.
Grimsby did not sit back and pulled themselves back into the tie thanks to Green’s impressive header at the near post.
Replays showed that Kabia appeared to make contact with the ball with his elbow before it went in, however the incident was not pulled up.
Kabia was in the mix again as he sent them level on aggregate just five minutes later, and his goal stood despite replays showing that he may have made his run beyond the Salford defence too soon as the ball was played into him.
And so extra time beckoned, but with little to separate the sides and with penalties fast approaching, the ball fell kindly for Cesay, who managed to beat the Grimsby back line, race into the box and finish coolly to send them through.
Cesay has now scored in successive games after drawing his side level against Grimsby in the first leg, having spent much of the latter half of the season out through injury.