Bish raced into a three-goal lead at half-time with goals from Nick May (2) and Jordan Annett and they held out despite Elvington pulling two back in the second half.
Strensall were due to host Barmby Moor but, with their pitch waterlogged, they made the trip to East Yorkshire and won 3-1.
The led 2-1 at half-time with goals from Mick Pipe and Jack Wilson, and Pipe managed a second goal after the break to ensure Strensall’s passage into the next round.
Fulford Reserves also suffered spot kick agony as they exited 3-2 on penalties after drawing 2-2 at Tadcaster Magnets Reserves.
The Tadcaster club’s first-team won 2-1 at York RI Reserves to book their quarter-final spot.
A Matty Bilham goal was enough to see Crayke Reserves beat Cliffe’s first team, although Cliffe almost forced extra time when they hit the bar in the last minute.
The postponed Junior Cup game between Selby Town Reserves and Sporting Knavesmire has been rearranged for Saturday, December 8.
The winners of that tie will host Bishopthorpe United in round four.
Osbaldwick have been drawn at home to Strensall, while Dunnington Reserves entertain Tadcaster Magnets and the Magnets second string are home to Crayke Reserves.