Mellor's strike proved the difference against a Pools side which has now lost seven successive home games in all competitions.
The hosts should have been awarded a penalty after 16 minutes when Jamie McCombe handled in the area from a Peter Hartley header but referee Jock Waugh played on.
North End, winless in the league since September 27, then scored the only goal five minutes later. Mellor picked up a loose ball and drilled a low, first-time shot into the net from 25 yards.
Lilywhites goalkeeper Thorsten Stuckmann saved from Antony Sweeney and James Poole as the home side searched for a leveller. The closest they came to scoring was on the hour mark when Poole galloped forward 70 yards in possession and whipped in a low shot that Stuckmann did well to push around the post.
Pools have only scored one goal - a penalty - at home since September 17, a run of 576 minutes.