On March 14, 2016, Jones attended LSU's pro day and chose to perform positional drills and all of the combine drills, but opted to stand on his broad jump performance. All together, he performed the 40-yard dash (4.39s), 20-yard dash (2.52s), 10-yard dash (1.52s), bench press (16 reps), vertical jump (35½"), short shuttle (4.26s), and three-cone drill (7.12s). He had an excellent overall performance in the combine drills and positional drills and his 40-yard dash (4.39s) would've finished first among all linebackers at the NFL combine. His time in the 40 was adjusted to an official time of 4.40s and dramatically increased his draft stock and profile. His pro day performance was described as "off the charts" by NFL analyst Gil Brandt. He attended private workouts and visits with multiple teams, including the New Orleans Saints, Indianapolis Colts, Philadelphia Eagles, New York Jets, and San Francisco 49ers. At the conclusion of the pre-draft process, Jones was projected to be a second or third round pick by NFL draft experts and scouts. He was able to raise his draft stock with his performances throughout the draft process and went from a projected third or fourth round pick to an all but guaranteed second rounder. He was ranked the sixth best outside linebacker prospect in the draft by NFLDraftScout.com, was ranked the sixth best linebacker in the draft by NFL analyst Mike Mayock, and was ranked the ninth best linebacker by Sports Illustrated.
The Atlanta Falcons selected Jones in the second round (52nd overall) of the 2016 NFL Draft. He was the eighth linebacker selected in 2016 and was the first of two linebackers the Falcons selected in 2016, along with fourth round pick (115th overall) De'Vondre Campbell. Jones was also reunited with LSU teammate Jalen Collins. The Falcons decided to draft Jones after linebackers coach Jeff Ulbrich scouted him at the combine. Ulbrich and head coach Dan Quinn thought Jones was an ideal candidate to play middle linebacker with his speed and pass coverage ability.
On May 5, 2016, the Falcons signed Jones to a four-year, $4.54 million contract that includes $2.16 million guaranteed and a signing bonus of $1.50 million.
Jones made his NFL debut and first NFL start in the season-opener against the Tampa Bay Buccaneers and recorded six combined tackles in a 31–24 loss. On September 26, 2016, Jones collected nine combined tackles, broke up two passes, and returned his first NFL interception for a 90-yard touchdown during a 45–32 road victory against the New Orleans Saints. His interception came in the fourth quarter off a pass attempt by Drew Brees and marked his first NFL touchdown. Jones was inactive for the Falcons' Week 5 victory at the Denver Broncos, after sustaining an ankle injury the previous week. During Week 14, he made five combined tackles, deflected two passes, and returned an interception by Jared Goff for a 33-yard touchdown during a 42–14 road victory against the Los Angeles Rams. Two weeks later, Jones recorded a season-high 13 combined tackles and broke up a pass during a 33–16 road victory against the Carolina Panthers.