The game was only four minutes old when Pier took the lead in their cup clash that was switched to their Sherburn base, having been called off three times at the back end of 2017.
Martin Cooper's cross into the box induced an error from the Catterick keeper and Tommy Barker reacted quickest to tap in.
It took until just after the half-time interval for Pier to double their lead, Sam Garnett flying down the right flank before whipping in a low cross that was turned into the net by a Catterick defender.
Pier finished their scoring in the 65th minute, Garnett again involved as he teed up Gary Thomas, who curled into the top corner.
Garnett was named as Pier's man of the match.
Itis Itis Rovers were dumped out of the same competition as they fell to a 5-0 defeat at home to Tockwith.
Player-boss Mikey Barker was left unimpressed by his side's performance, although he singled out Ali Jones as his side's star man on a disappointing day for the club.
Division One leaders Edgehill hammered Flamborough 11-2 to continue their march towards the title.
Debutant Tyson Stubbings, who signed this week from Goal Sports, banged in a hat-trick on debut, although he could've netted four as he missed a penalty.
Man of the match Jamie Patterson hit a double, while Edgehill's other goals came from Kieran Link, Jackson Jowett, Luke Jones, Jamie Stewart and Danny Collins.
Clegg was left delighted by his side's performance. He said: "It was a very dominant performance on a heavy pitch. Tyson (Stubbings) had a good debut, scoring a pen and missing a pen before going on to get a hat-trick."
Seamer stayed in second spot as they picked up a deserved 2-0 win from their trip to take on Filey Town.
League top-scorer Danny Glendinning put Seamer ahead after half-an-hour before Ali Caw found the bottom corner with a rasping drive on the hour-mark to wrap up their win.
Midfielder Dean Craig was Seamer's star man, while Caw also shone.
Ellery Liley was Filey's man of the match.
Hunmanby United picked up their first win of the season after they beat Newlands 2-1 to leap off the foot of the table.
Newlands led at the break, Nick David firing in the only goal of the first half.
The hosts levelled matters up with 15 minutes left on the clock, Cameron Dobson latching onto a ball over the top and beating the keeper to the ball before poking on for sub Robbie Harrison to tap in.
United sealed the win as they took a 2-1 lead five minutes later, Ollie Milner feeding James Pinder to tap in at the back post.
Hunmanby boss Leigh Franks said: "We deserved to win the game, the whole defence played well but Ben Briggs was our man of the match."
Jordan Mintoft and Jack McFarlane shone for Newlands.