The sides played out a goalless draw at the CNG Stadium on Saturday and will face-off once again at the Racecourse Ground tonight (Tuesday), 7.45pm kick-off.
"I think we've got to be confident, we've only lost once in 16, after all," Weaver said.
"There's a good mentality in the group and we don't expect to come unstuck, but we have every respect for the game and also for the opposition.
"They're a big club historically, they've got some good players and will be tough to break down, that's for sure. It should be a good FA Cup battle."
Town looked the more likely during Saturday's fourth qualifying round clash, forcing Dragons goalkeeper Rob Lainton into a couple of excellent saves, hitting the woodwork and passing up a glorious late opportunity to snatch victory when Mark Beck headed over the bar from close range.
"I thought we had good chances to score and some gilt-edged chances where we missed. It could have been a very different scoreline," Weaver added.
"I fancied him [Mark Beck] to score if I'm honest. It's a shame for Becky, but he'll score a lot of those this season, of that I've no doubt."
Having drawn 0-0 in the National League back in September, Town and Wrexham have played more than three hours of football already this season without a goal being scored.
And Weaver's troops know they will have their work cut out at the Racecourse Ground where their hosts have conceded just once in their last five outings.
The men from Wetherby Road do however boast the National League's best away record, with five wins and three draws to their name on the road this term.
Striker Dominic Knowles and centre-half Callum Howe both face fitness tests ahead of the match, but otherwise Town are expected to be at full-strength.
The winner of this evening's contest will travel to National League South outfit Weston-super-Mare AFC in the competition's first round on Saturday November 10.