The visitors started the game on top, and their bright start was rewarded with a goal on 11 minutes.
Moore was afforded time and space in the middle of the park to travel forward, before firing a pinpoint left-footed strike into the bottom corner past a helpless Gartside after a touch of fortune from a ricochet.
Carrick weren't behind for long and levelled midway through the first half through Heatley, who calmly passed the ball beyond Chris Johns after latching onto Jimmy Callacher's long headed clearance.
Matthew Snoddy came to the hosts' rescue minutes later, clearing Kitt Nelson's strike off the line, with Johns then saving another Heatley shot straight up the other end.
In an end-to-end first half, the Blues had the last say as Fitzpatrick found a way past Gartside at his near post to restore the visitors' lead before the break from outside the area.
The hosts levelled for the second time just seven minutes into the second half as Gibson rose to meet Heatley's cross to head past Johns who was rooted to his spot.
Shields had the opportunity to put his side in front for a third time from the penalty spot after Kym Nelson brought Rhys Brennan to ground, but Gartside tipped his penalty around the post at full stretch with a superb right-hand stop.