The Boro were superior in the opening exchanges and Jake Charles had an early effort tipped around the post by the goalkeeper
Charles was not to be denied at the second time of asking netting after a corner fell into his path.
The Boro continued to dominate the opening stages and Tom Tonks' long throw-in found the head of Theo Bailey-Jones who rattled the crossbar with his headed effort.
Bamber Bridge sent their hosts a warning sign when Rob Apter ran though on goal before forcing a save from the leg of goalkeeper Pilling on 55 minutes.
Apter was not to be denied ten minutes later with a free-kick effort to remember slamming his effort in off the underside of the crossbar.
Bamber Bridge completed the fightback when Apter saw his effort palmed by Pilling into the path of Sheldon Green to put his side ahead.