They travel to Glanford Park on Tuesday night, after a 0-0 draw in the original tie, with the winner hosting League Two side Wrexham in the second round.
A Cup run would provide a welcome boost to the players who have let their heads drop after an impressive start to the season.
"I believe in the players and John Ward believes in them," insisted Downing, who took charge of the first-team at Saltergate while manager John Ward recovered from a knee operation.
"After the first few games we were sixth in the table and the chances were going in but now that has stopped but I don't think we are that far away. The players have to start believing that so that we can start getting results.
"There are some good footballers in there, but the current run of results has affected their confidence.
"They have to be a bit braver and have the courage to get on the ball and try and play with a bit more quality. The players have the ability but it's a case of matching that with belief that they can do it."
