He earned the victory in his lone start of the season against Prairie View A&M and pitched three scoreless innings of relief against McNeese State.

He finished the year with a 3.28 ERA in 93+1⁄3 innings, limited opponents to just a .242 average, struck out 75 batters while walking just 26.

Finnegan was later named the starting pitcher for the west team in the 2014 Midwest League All-Star Game, where he earned the win tossing one shutout inning.

Receiving a promotion to the Midland RockHounds of the Double-A Texas League, he had a 0–1 record with an 11.81 earned run average.

Finnegan split the 2018 season between the rookie–level Arizona League Athletics, Midland, and Nashville, going 1–3 with a 4.84 ERA over 44+2⁄3 innings.

He split the 2019 season between Midland and the Las Vegas Aviators, going a combined 3–2 with a 2.31 ERA over 50+2⁄3 innings,[4][5][6] and was named a 2019 Texas League All-Star.

[9] He made his major league debut on July 25, 2020, against the New York Yankees, allowing one hit and no runs in one inning of relief.

On January 13, 2023, Finnegan agreed to a one-year, $2.3 million contract with the Nationals, avoiding salary arbitration.

[14] On July 15, 2024, Finnegan was added to the National League All-Star team as a replacement to the injured Ryan Helsley.