Sweeney had impressed for most of the game and he got the chance to mark his debut with an equaliser in the 83rd minute from the penalty spot. Unfortunately, his spot kick was well saved by Truro keeper Tom McHale, who had a fantastic day between the sticks.
Two goals either side of the penalty from Noah Keats was enough to sink The Beavers, who were aiming to get to the First Round Proper for the first time since 2007. Truro are currently on their best run in the competition, with their previous best being the Third Qualifying Round back in 2011.
Sweeney will be hoping to get the side back to winning way in their next fixture when they face Whitehawk away from home in the Vanarama National League South.
Justin Amaluzor played just under an hour for Hemel Hempstead Town, with Tom Day completing the 90 minutes in a 2-0 win away at Eastbourne Borough.
Joe Payne played no part in Solihull Moors’ 1-1 draw with Ossett Town in the Fourth Qualifying Round of the FA Cup.