North End – relegated last season and beaten 4-2 at home by Colchester in their League One opener on Saturday – led when Matt Tootle put through his own net.
But Crewe hit back through Shuan Miller and Dave Artell to lead 2-1 at the break before late goals from Neil Mellor and Hume sealed the win.
Preston led inside the third minute when Crewe defender Tootle diverted into his own net from six yards out.
Yet the response from Crewe was impressive and they equalised in the 18th minute when Miller drove a left-foot shot into the bottom corner from 12 yards out.
And the visitors forged ahead five minutes before the break when Nick Powell crossed dangerously from the left flank and Artell headed into the net from the edge of the six-yard box.
Mellor then fired home from 25 yards to level the scores before Hume headed home a cross from Danny Mayor.