It was a fourth successive victory in all competitions for the in-form Latics.
Shrewsbury started brightly and striker Gozie Ugwu, on loan from Reading, fired them in front in the 16th minute with his first goal for the club.
Ugwu made the most of some poor defending to sneak in and drive a shot from just inside the penalty area into the ground, which looped up and over goalkeeper Mark Oxley.
But Oldham responded well and were full value for their equaliser on the stroke of half-time.
But the equaliser eventually arrived in the 44th minute when Charlie MacDonald ran on to a neat through-ball from Anton Rodgers to fire under advancing keeper Chris Weale.
An open second half saw chances at both ends, but Philliskirk won it for Oldham when he finished clinically after Darren Jones had been dispossessed by MacDonald.
Tom Bradshaw nearly grabbed a late equaliser, but was denied by the fingertips of Latics keeper Oxley.