As a freshman, Elliott was named the starting kicker after summer training. In his 2013 season, Elliott connected on 16 of 18 field goals. After his freshman season, he led the Tigers with 72 points scored (24/24 extra points and 16/18 field goals). He set a Memphis record with his 56-yard field goal at USF, breaking the 2005 record of 53 yards set by Stephen Gostkowski. He was named the American Athletic Conference Special Teams Player of the Week and also one of three players to be named "Stars of the Week" by Lou Groza Award. In Elliott's sophomore year, he led the American Athletic Conference by averaging 9.2 points per game. For the second year in a row, he was named First Team All-Conference and Conference Special Teams Player of the Year. He scored 120 points during his season, making 21 of 32 field goal attempts and converting all 57 extra-point attempts. His 21 successful field goals, 57 extra points, and 120 total points scored led the American Conference. During the 2014 Miami Beach Bowl, Elliott kicked the fourth-longest field goal in bowl game history, a 54-yard attempt that sent the game into a second overtime and led to a 55-48 victory. In the 2015 season, he converted all 63 extra point attempts and 23 of 28 field goal attempts for a team-leading 132 points scored. His 63 extra points and 23 field goals led the conference in the 2015 season. In the 2016 season, he converted all 58 extra point attempts and 21 of 26 field goal attempts for a team-leading 121 points. His 21 field goals tied him with Tulsa's Redford Jones for the conference lead. In his collegiate career, he converted all 202 extra point attempts and 81 of 104 field goal attempts. At the end of the 2016 season, his successful collegiate career ranked him first in total points, extra points, and field goals while also finishing third in field goal percentage in conference history.

On September 12, 2017, the Philadelphia Eagles signed Elliott off the Bengals' practice squad after the Eagles' starting kicker, Caleb Sturgis, was placed on injured reserve. Elliott made his first NFL appearance in Week 2 of the 2017 season against the Kansas City Chiefs. In the game, he converted both extra point attempts and two out of three field goal attempts. On September 24, 2017, at Lincoln Financial Field in Philadelphia, Elliott kicked a 61-yard field goal to beat the New York Giants on the final play of the game, resulting in a 27–24 victory, earning him NFC Special Teams Player of the Week. It was tied for the 7th-longest field goal in NFL history, and the longest since November 2015. It was also the longest field goal in Eagles history, the longest ever kicked in Philadelphia in an NFL game, and the longest ever by an NFL rookie. In Week 4, against the Los Angeles Chargers, he put together a solid performance in converting all four field goal opportunities and both extra point attempts for a season-high 14 points scored. During the Week 11 game against the Dallas Cowboys, Elliott made a tackle during the starting kick return. He would later be pulled out of the game for a head injury and would not return. The Eagles finished with a 13–3 record, won the NFC East, and earned a first-round bye. Against the Atlanta Falcons in the Divisional Round, he missed an extra point but converted three field goal attempts in the 15–10 victory. In the NFC Championship against the Minnesota Vikings, he converted all five extra point attempts and one field goal attempt in the 38–7 victory. During Super Bowl LII on February 4, 2018, Elliott made all three of his field goal attempts, including a 46-yard field goal in the fourth quarter that helped seal the Eagles' first Super Bowl victory over the New England Patriots.