The first half produced very few chances up either end, with Kitt Nelson having the best opportunity for the hosts. His strike was blocked inside the area just as he was unleashing a strike following Adam Frizzell's pinpoint through ball.
For the visitors Tiarnan Mulvenna had their best chance, with Chris Shields coming to Linfield's rescue with a vital block just inside the box.
The Blues began the second half firmly on the front foot but struggled to produce a clear chance as the Seasiders stood firm.
Lee Feeney's side were then reduced to 10 men after 54 minutes as Lynch was sent off for a late challenge on Dane McCullough, with referee Declan Hassan producing the red card without hesitation.
Three minutes later and the Blues hit the front through Moore, who tucked Rhys Brennan's inviting cross from the right into the bottom corner.
David Healy's side were then awarded a penalty as Steven McCullough wrestled Adam Frizzell to ground inside the box, with Shields stepping up to double their lead on 65 minutes.