The 18-year-old joins the club for an undisclosed fee and has signed a four-year deal with the Whites, running until the summer of 2024.
Gelhardt made his Wigan debut in August 2018 at the age of 16 against Rotherham United in the Carabao Cup.
He made one further appearance for the Latics during the 2018/19 campaign, earning a league debut in the Sky Bet Championship against Birmingham City.
Last season the striker featured more regularly for the DW Stadium outfit, making two starts and 17 outings from the bench in all competitions.
He scored his first goal for the club in a 2-2 draw with Hull City in the Championship last September.
The striker has been capped at England at Under-16, Under-17 and Under 18 level and featured in the European Championship in 2019 for the Under-17s.