He played college football for the Ole Miss Rebels, and was selected by the Indianapolis Colts in the third round of the 2014 NFL draft.

In addition to football, Moncrief also competed in track & field for the Lions, where he was one of the state's top performers in the long jump.

He was a two-time 3A state champion in the long jump event, with a personal-best mark of 7.76 meters (25-1.5).

[5] Moncrief announced on January 3, 2014 that he would forgo his senior season and enter the 2014 NFL draft.

[10] On October 26, 2014, in a game vs. the Pittsburgh Steelers, Moncrief scored his first NFL touchdown on a pass from quarterback Andrew Luck.

In Week 13, Moncrief had 3 receptions for a career-high 134 yards and 2 touchdowns to help the Colts beat the Washington Redskins 49–27.

[12] In 2016, Moncrief missed six games with a shoulder sprain and another with a hamstring issue but still managed to catch seven touchdowns.

[14][15] On March 15, 2018, Moncrief signed a one-year, $9.6 million fully guaranteed contract with the Jacksonville Jaguars.