Stroud began his college football career at Ohio State University, where he redshirted as a backup to Justin Fields during his true freshman year in 2020. In that year, Stroud played just one total snap and scored a 48-yard rushing touchdown.

In 2021, following Fields' departure to the 2021 NFL Draft, Stroud was named the starting quarterback for Ohio State as a redshirt freshman. He beat out freshmen Kyle McCord, Quinn Ewers, and fellow redshirt-freshman Jack Miller III for the starting spot. Coach Ryan Day praised Stroud's decision-making, leadership skills, and accuracy as the reasons for his selection.

Stroud started every game for the Buckeyes in the 2021 season aside from a Week 2 game against Akron, where he sat out to rest a shoulder injury sustained in the season opener. During the season, Stroud earned first team All-Big Ten Honors, was named the Big Ten quarterback of the year, won Big Ten Freshman of the Week six times, and was a finalist for both the Davey O'Brien Award and the Heisman Trophy (losing both to Bryce Young).

In the 2022 season, Stroud was once again a standout player and one of the best in college football. On October 8, 2022, Stroud threw 6 touchdown passes against Michigan State, setting a conference record for "most 6 passing touchdown games in a career" with 3, and passing Justin Fields to move to second place on the Ohio State career passing touchdowns list. Despite coming up short against rival Michigan, Stroud and the Buckeyes were selected as the fourth and final team for the College Football Playoff. In the Peach Bowl semifinal against the top-seeded Georgia Bulldogs, Stroud had a strong performance, throwing for 348 yards and 4 touchdowns, but the Buckeyes fell 42-41, ending their season.

At the conclusion of the 2022 regular season, Stroud was named a finalist for the Heisman trophy for the second year in a row. He finished third in the voting behind winner Caleb Williams from USC and Max Duggan from TCU. On January 16, 2023, Stroud announced that he would forgo his remaining two years of college eligibility and enter the 2023 NFL Draft.