Despite sitting 14th in the division below, Carterton came into the tie on a strong run of five straight wins -- including an impressive 3-0 victory over promotion-chasing Redditch Borough.
A running start from Pershore saw a quick goal 90 seconds into the match when a pinpoint Macca King cross was headed against the upright by Cal Harper. Jack Griffiths-Perry secured a goal from the rebound.
Building on their early lead, the team continued to press the opposition but got into foul play moments after the first goal.
A controversial penalty kick plea saw a break-away Harper get bundled over from behind by a Carterton defender.
Appeals for a penalty and a red card were waved away by the referee.
The visitors saw their efforts rewarded, however, when on 17 minutes a clearance from Cian Sheehan was flicked on by Griffiths-Perry, sending Joel Hill through on goal, and the striker finished emphatically from the left side of the area.
Heavy rain made play difficult for both teams, with some players slipping on the pitch.
However, Carterton found a way back into the game on 22 minutes with a short corner routine and a clipped cross into the box allowing them to secure their only goal of the match.
Pershore's goalkeeper Joe Cunneen saw some action after having to parry a long-shot effort from Carterton before it was knocked wide of the goal.
With the scoreline at 2-1, Carterton began to find their footing.
Some efforts from Jude Hill and Jack Griffiths-Perry went without reward as the team pressed to improve on their lead.
After the first half of play, the visitors continued on the front foot.
Passed down the wing, the ball was crossed into the box by Griffiths-Perry to Cian Sheehan -- after Reece Day had sent Griffiths-Perry down the wing -- who in turn struck the ball against the post on the 52nd minute.
Another spell of maintained possession for Carterton threatened the Pershore defence but failed to convert into any goals.
Ty Griffin forced another save from the Carterton keeper before Pershore widened the gap at the 67th minute.
Reece Day secured a third goal after picking up a ball just after a Carterton corner which failed to be fully cleared.
Down 3-1, Carterton saw its keeper in the late minutes of play be sin-binned for 'dissent', leaving an outfield player to take over between the goalposts for the closing minutes.
With an extra man Pershore swept forward, with substitute Archie Thurston securing the team its fourth goal after the replacement keeper's clearance fell to Kyle Clayton, who combined with Lucas Colley and Jude Hill before Colley's cross was swept home by Thurston.
In stoppage time, there were still chances -- Colley fired wide, Isaac Nharo saw a shot blocked after good work from Griffiths-Perry, and Griffiths-Perry himself had the final effort of the match -- leaving the scoreline at 4-1.
Pershore is now set to face Hereford Pegasus in the next round.