Altrincham's chairman said they should be awarded the win after Colwyn Bay turned down a replacement from the crowd, claiming he had been drinking.
An offer from Robins coach Ian Senior to run the line in the FA Trophy tie at Colwyn Bay was also rejected.
"I hope common sense prevails," Alty chairman Grahame Rowley told BBC Sport.
"We hope we don't have to come back for seven minutes. We were dominating. They should award the game to us."
Colwyn Bay expect the game to be replayed in full on Tuesday, and manager Frank Sinclair told BBC Radio 5 live's Non League Show: "You can't just have any Tom, Dick or Harry come in and finish the last 10 minutes of the game.
"There was no-one on our side that had any officials' badges. One of their staff said he would see the rest of the game out with no qualifications. It's human nature that if there was a big decision to make, we wouldn't have got it.
"I've been within my rights to not agree to carry on."
And at first glance the rule book seems to be in Colwyn Bay's favour:
FA Trophy rules
"When a first match has been postponed, or is abandoned before the completion of 90 minutes and neither club being at fault, it shall be played on the same ground on or before the following Thursday.
Matches abandoned through the fault of the clubs shall be dealt with by the National Game Board."
Altrincham must now wait for the Football Association to decide whether the tie must be replayed, the remaining seven minutes played, or whether Altrincham will be awarded the tie.