The Hoops came closest in injury-time when Gary Hooper saw a shot cleared off the line before a penalty claim was denied.
Neil Lennon handed Adam Matthews and Kelvin Wilson their first taste of action for the club as his team failed to find a cutting edge despite superior possession.
Former Cardiff full-back Matthews put in an accomplished display in the first half and unleashed a powerful long throw on several occasions, while Wilson made some decent interventions.
Lennon left out Thomas Rogne and Charlie Mulgrew as a precaution after they reported stiff muscles, while Kris Commons remained on the bench for the same reason. Daniel Majstorovic, Georgios Samaras, Cha Du-ri and Ki Sung-yueng are still to join up with their team-mates following international commitments, while Emilio Izaguirre is recovering from a hamstring injury.
Celtic carved out the first chance when Mark Wilson intercepted a pass and slipped the ball through for Daryl Murphy, but the striker's touch allowed the goalkeeper to smother.
Murphy was soon handed another chance after a clever pass over the top from Beram Kayal allowed James Forrest to cross, but the former Sunderland striker glanced the ball wide.
The Bhoys moved the ball about quickly without breaking last season's A-League Grand Final runners-up down, and Mariners came into the game towards the break. Kelvin Wilson did well to block a strike from Mariners newcomer Adriano Pellegrino, who was linked with Celtic last year before having a trial with Motherwell.
The visitors stepped up a gear immediately after the break without carving out a real chance, but Mariners settled down and Celtic soon had a big escape when Pellegrino was off target with a free header from a good position. Forrest fired a yard over from just outside the box, but Hearfield then found the top corner after Ledley failed to deal with a low cross from the left.
Ledley almost made amends when he flicked the ball into Hooper's path but the striker's finish was cleared off the line. Anthony Stokes played the ball back in before claiming for handball and caught the ball in expectation of a penalty, but none was forthcoming.