Some Preston supporters ran on to the pitch at the final whistle
It was an unsavoury end to the first meeting between the two rivals since February 2010.
That previous encounter - a Championship fixture - finished goalless and there was little during the first half at Deepdale on Monday to suggest their latest meeting would be much different.
Pool keeper Gilks made a comfortable save from a Holmes free-kick in the opening minutes but had little else to do before the break.
The visiting side - with summer signing Michael Chopra making his full debut as one of three changes from the side that won at Doncaster on Saturday - looked most threatening when Tom Ince was in possession.
Ince angry at Preston pitch invasion
The 21-year-old shot wide midway through the opening half and later showed the North End defence a clean pair of heels before drilling a left-foot strike wide of Declan Rudd's goal.
The tie opened up after the break, with a more assured and attacking North End going close with a shot across goal from the inventive Hume and, later, a low long-range strike from Holmes, both of which were saved by Gilks.
Angel Martinez, Kirk Broadfoot and Chopra all threatened with headers for Blackpool - with the last effort bouncing up off the turf and narrowly over Rudd's crossbar.
But the tie swung decisively in the League One side's favour when Clarke scored North End's first goal of the season as boss Simon Grayson saw his team win against the side that he managed from 2005 until he left for Leeds United in 2008.
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