Debutant Philliskirk fired home on the turn to give Oldham a 10th-minute lead and then struck again just before the half-hour mark when he converted a cross from left-back Joseph Mills.
Oldham then moved further clear when Adam Rooney powered in a near-post header from Matteo Lanzoni's cross from the right.
Shrewsbury were not without their chances as Latics goalkeeper Mark Oxley saved well from Curtis Main, Jon Taylor and Aaron Wildig.
Oxley was beaten by the power of Graham Burke's 20-yard drive, which gave Shrewsbury some hope of a comeback just before the interval.
But Oldham continued to look dangerous in the second half and lively Dutch winger Sidney Schmeltz fired in a 75th-minute fourth after cutting in from the left and beating 'keeper Joe Anyon for pace.