In a game that saw two players from each side sent off late on, Rovers dominated the first half to lead 3-0 thanks to a Danny Franks’ brace of goals and a strike from Lee Bunce.
Hemingbrough pulled a goal back thorugh Luke Zacharenko but Rovers made sure of their quarter-final success when Chris Dixon and Chris Dyson struck to confirm a 5-1 win.
Britannia Tadcaster and Tadcaster Magnets are both through to the semi-finals of the West Riding County Cup.
Britannia were 1-0 winners over Premier Leeds League side Queenswood Crusaders. Leading marksman Adam Field bagged the crucial lone goal in the second half.
Magnets won their tie 2-1 with Matty Howgate and Ben Wilson sharing the plunder.