He played college football for the Eastern Michigan Eagles and was selected by the Oakland Raiders in the fourth round of the 2019 NFL draft.

[19] In week 17 against the Broncos, Crosby made 1.5 sacks on fellow rookie Drew Lock in a 16–15 Raiders loss.

[20] Crosby finished his rookie season with ten sacks, 47 total tackles, four passes defended, and four forced fumbles.

[23] In Week 3 against the New England Patriots, Crosby recorded his first two sacks of the season on Cam Newton during the 36–20 loss.

[24] In Week 5 against the Kansas City Chiefs, Crosby recorded his first sack on Super Bowl MVP Patrick Mahomes in a Raiders 40–32 win.

[31] Crosby would set a number of personal bests that season, which included passes defended (7) and a team-leading amount of quarterback hits (30), more than doubling his career highs at that point for both statistics.

[45] Crosby was named AFC Defensive Player of the Week for his two-sack game against the Baltimore Ravens in a 26–23 win.

He was ruled out ahead of the Raiders' Week 15 matchup against the Atlanta Falcons and on December 14, it was announced that he plans to undergo surgery for the injury, likely ending his season.

Crosby totaled 7.5 sacks and 20 quarterback hits this season, in addition to being tied with T. J. Watt for the league lead with 17 tackles for loss.

[51] In 2023, Crosby tattooed his chest with the words "Be Legendary" with pictures of Muhammad Ali, Kobe Bryant, Michael Jordan and an image of his daughter as an infant.