Turning out for Brian Stein's men were three trialists, former Manchester City right back Chris Ramsey, Watford left-back Louis Lavers and ex-QPR winger Chris Arthur.
But Enoch Showunmi made no mistake when Tom Elliott's shot was pushed into his path by Grimsby goalkeeper Leigh Overton.
Leeds could have doubled their advantage from the spot on 40 minutes. David Prutton had picked out Showumni with an inch-perfect pass but he was bundled over by Lavers as he prepared to shoot.
Lavers was booked but Elliott clipped his penalty down the centre of the goal and saw it bounce off the crossbar.
Leeds continued to control the match in the second period and a shot from Robinson came back off the woodwork, but Grimsby's chance came at the death when United keeper Casper Ankergren was penalised for handling the ball outside his area.
Heywood scored in added time when, after the free-kick from striker Chris Jones, his first two efforts were blocked before the third attempt went in off the underside of the bar.
Leeds United: Ankergren, Darville (Milne 81), Michalik, Huntington, Sheehan, Prutton, Falkingham, Kilkenny (Hatfield 63), Robinson, Elliott, Showunmi (Whitwell 65).
Sub Unused: R Jones.