He played college football for the UTSA Roadrunners and was selected by the New Orleans Saints in the first round of the 2018 NFL draft.
[4][5] As a senior in 2017, he was the Conference USA Defensive Player of the Year after recording 55 tackles and 8.5 sacks.
[17] In a Week 2 matchup against the Los Angeles Rams in 2019, Davenport sacked Jared Goff as the Saints lost 27–9.
[18] In a Week 5 game against the Tampa Bay Buccaneers, Davenport sacked Jameis Winston twice in the 31–24 win.
[22] In Week 7 against the Carolina Panthers in 2020, Davenport recorded his first sack of the season on former Saints' teammate Teddy Bridgewater during the 27–24 win.
[27] On June 15, 2022, Davenport announced he had undergone five total offseason surgeries: two on his shoulder and three on his left hand.
[32] In a game at the Arizona Cardinals on September 22, Davenport suffered a season-ending triceps injury.