Redcar, who currently ply their trade in the Teesside League, will be a tough nut to crack, especially on their home patch.
But with Falcons high on confidence coming into the contest, it promises to be a cracker of a contest.
Peirse's side will now put their Harrogate Premier League campaign on the backburner for two weeks with tomorrow's cup clash followed by their mini-series against Harlow Hill, who they will welcome next Saturday in the Challenge Cup.
"One or two of the players know about Redcar, they are a Teesside League side and they'll be a strong, physical side," said Peirse.
"But I'll have a full strength team to pick from. Adam Ryan's back, Les Williams is in, so they'll be no excuses.
"We know if we want to win the North Riding Cup we've got to be beating teams like Redcar, so we'll be going out to win the game.
"I'm fairly confident, it's a game we could and should be winning but football being football, you never know."
Thirsk come into the contest on the back of Saturday's 8-1 thrashing of Harlow Hill, a result that moves them six points clear at the top of the table with Pateley Bridge's proposed fixture postponed.
However, the table-toppers did not have it all their own way for much of the contest with home side Harlow dominant early in the match.
"They started the game really well and it looked like they were the team at the top of the table and we were the team near the bottom," continued Peirse.
"But we responded really well after that.
"You could see their heads all go down when we got the equaliser and from then on there was only going to be one winner."
Thirsk's leveller was scored by James Robson and they added three more before the break, through Scot Clark (2) and Bruce Cree.
In the second half Mark Cobb scored a well taken hat-trick and the Harlow 'keeper, who had a blinder all afternoon, punched into his own net to finish the rout.
"We completely dominated in the second half," added Peirse.
"Once we levelled there only looked like there was going to be one winner. Their 'keeper had a blinder and made four or five superb saves. He scored an own goal but that was a bit unfair on him.
"It keeps the run going for us and now we're six-points clear at the top of the league. Pateley Bridge were called off on Saturday and a few of their players came down to watch us.
"All the pressure is on them to win their games in hand."
Thirsk now have one league game remaining before Christmas with Peirse and his men looking to go out with a bang at Burley Trojans.
On January 3 they gear up for a tasty local derby against rejuventated Bedale Town Reserves at Flatts Field.
Peirse is staying wary before the big game, especially after Bedale's thumping victory over Eccleshill United at the weekend.
"They easily beat Eccleshill and although they're not a good side you've still got to do it," added Peirse.