Elliott confirmed the move on social media.
Fly Eagles Fly!— Jake Elliott (@jake_elliott22) September 12, 2017
A rookie out of Memphis, Elliott converted 78 percent of his field goal tries over four seasons at the collegiate level and holds American Athletic Conference records for career points (445) and made field goals (81).
Sturgis suffered a hip flexor injury on the opening kick Sunday against the Washington Redskins and is expected to be out four weeks, a source told ESPN's Adam Schefter.
The Bengals selected Elliott with one of their fifth-round picks after struggling to fix their kicking game in 2016. Longtime kicker Mike Nugent was released late in the season after he missed six field goals and six PAT attempts. Randy Bullock, Nugent's replacement, missed a potential game-winning kick against the Texans in December.
Elliott displayed a strong leg on field goals and kickoffs, just missing a 60-yard attempt wide in the Bengals' final preseason game. Bullock, however, had an almost perfect camp. He did not miss a kick in a preseason game until the final week and was also more consistent in practice.
The Bengals highly value their draft picks and rarely cut rookie draftees in their first year. That made waiving Elliott a difficult decision, because they knew there was a strong possibility he would be picked up by another team. But Bengals coach Marvin Lewis said Bullock was the clear winner of the job.
"There's a lot of data, a lot more data than you're aware of," Lewis said. "So we have to go by that or else there's no reason to keep track of everything we do. Obviously if it was anything close, Jake would have won the job."
ESPN's Katherine Terrell contributed to this report.