Elliott attended Waco High School, where he practiced football and soccer. In soccer, he was a four-time All-district selection. He walked on at Texas Tech University. As a sophomore, he was named the starter at placekicker, making 9 out of 19 field goal attempts. As a junior, he made 14 out of 17 field goal attempts for a school record 82.4%. As a senior, he made 17 out of 26 field goal attempts, set the school's consecutive extra point record and was second in the conference with 85 points. Against Cal State Fullerton University, he tied the school record for field goals in a game with 4, including a career-long tying 52-yarder. Elliott finished as the school's All-time leading scorer at the time, while making 40 out of 62 field goal attempts and 100 out of 101 extra points.. In 2015, he was inducted into the Texas Tech Athletic Hall of Fame.
Elliott was signed as an undrafted free agent by the Dallas Cowboys after the 1992 NFL Draft, to compete as the replacement for kicker Ken Willis. He recovered from a pulled groin muscle in training camp, to be named the starter over Brad Daluiso. He started the regular season slowly until hitting 13 straight field goals and registering 27 touchbacks, which at the time were both franchise records. He set a franchise record for points by a rookie, his 24 field goals were third in club history, his 119 points ranked fourth in the NFL and led all rookies. He contributed to the team winning Super Bowl XXVII. On September 14, 1993, he was waived after missing two field goals in a 13–10 loss against the Buffalo Bills in the second game, contributing to an 0–2 start. He was replaced with Eddie Murray.
On April 7, 1994, he signed as a free agent with the Kansas City Chiefs, after Nick Lowery was released in a salary-cap move. He registered 25 of 30 field goals. In 1995, he made 24 of 30 field goals, contributing to the team going 13–3 in the regular season and clinching home field advantage throughout the playoffs, making them heavy favorites. But he is best remembered for missing three field goal attempts from 35, 39, and 42 yards in a to the Indianapolis Colts in January 1996. He suffered a late-season swoon, as his final miss against the Colts was his ninth missed kick in five games. On February 16, 1996, the Chiefs decided not to make him a qualifying offer as a restricted free agent, effectively releasing him, even though he had an 81.7% field goal accuracy during his two seasons, ranking as the second-most accurate kicker in Chiefs history.
