Littlehamton were faultless from the penalty spot as they booked their place in the next round.
A ball in from Dan Perry looped and hit the top of the crossbar on six minutes as Hill went close early.
The game had little quality from either side, Will Huffer made a fabulous save to deny a certain headed opener for the hosts while Harry Lawson spurned Hill’s best chance thanks to keeper James Binfield.
The second half didn’t spark into life up until Marigolds captain Jordan Layton was sent off for a late tackle on Harry Lawson.
Hill nearly took the lead when Ollie Davies flicked a cross on but Matt Astle denied him. The visitors should have gone ahead as sub Marcel Powell takes a touch but smashed wide when he should have done better.
The Golds’ joint manager George Gaskin was then sent off for an elbow on Nathan Cooper.
The last chance came and went when Ollie Davies hit the underside of the bar Dan Perry was the unfortunate man to miss from the spot to send the home side through.
Hillians: Will Huffer, Bobby Price (Reggie Ward 62), Hamish Morrison, Brannon O’Neill, Nathan Cooper (Alex Malins 90), Elliott Bresciani, Ollie Davies, Kieran Rowe, Dan Perry, Harry Lawson (Marcel Powell 68), Joe Overy (Alex Brewer 84). Subs not used: Sam Forbes-Parker