Davis double earns newly promoted Garforth their first NPL win for over a decade.
Liversedge almost led at the Bannister Prentice Stadium in the first minute, but Joe Walton headed wide from close range.
Sam Barker forced the first save of the game soon after.
The visitors had the majority of the first half, but Matt Antcliff chipped Garforth into a lead five minutes from the break, beating Sedge keeper Parkin to a through ball.
A much improved Miners grasped the lead through a stunning thirty yard strike by substitute Jay Davis ten minutes from time.
Davis then sealed Garforths winning return to the NPL with a composed finish in the ninetieth minute.