He played college football for the Arkansas Razorbacks and was selected by the Jacksonville Jaguars in the sixth round of the 2016 NFL draft.

[4][5] He passed for 1,552 yards, completing 49.6 percent of his throws, and threw for 13 touchdowns and 10 interceptions for a team that finished 3–9 under first year head coach Bret Bielema.

[16] Allen was the Jaguars' third-string quarterback his rookie season behind Blake Bortles and Chad Henne.

[18] He was inactive for every game of the season as the Rams' third-string quarterback behind Jared Goff and Sean Mannion.

[27][28] He was promoted to the active roster on November 23, 2020, after starting quarterback Joe Burrow suffered a season-ending injury.

[29] On November 25, 2020, the team announced that Allen would start for the Bengals during their Week 12 game against the New York Giants in place of the injured Joe Burrow.

[31] Allen started the final game for the season against the Baltimore Ravens, where he completed six of 21 passes for 48 yards, two interceptions, and a passer rating of 0.0 as the Bengals lost 38–3.

In Week 9, with the game out of reach for the Bengals, Allen would go 1 for 2 for 6 yards passing, while rushing once in the 41–16 loss to the Browns.

[38] Allen was subbed in at the end of the Week 9 game against the Carolina Panthers due to the Bengals resting their starters for the majority of the fourth quarter.

[41] He began the 2024 season as the second-string quarterback behind Brock Purdy after beating out Joshua Dobbs during training camp.

[45] He has a younger brother, Austin Allen, who became the starting quarterback for the Razorbacks after Brandon completed his college eligibility.

Austin signed with the Tampa Bay Buccaneers as an undrafted free agent following the 2018 NFL draft.