Kaelin Clay put the Ravens ahead in the first quarter when he collected a punt and beat several Browns tacklers to race down the sideline and record an 82-yard touchdown.
© Getty Images
However, Coons then made an error from the kickoff by sending his kick out of bounds down the sideline, allowing the visitors to start on their 40-yard line.
The Browns rallied as Josh McCown evaded the pass-rush and threw on the run to find Marlon Moore for a three-yard score, while Coons reduced the arrears to four points with his second field goal before the break.
In the second half, Schaub committed his first mistake of the match when he was picked off by Karlos Dansby, who returned the ball 52 yards for a touchdown to hand the home side the lead for the first time in the game.
The Ravens quarterback managed to atone for his error by leading a go-ahead touchdown drive when he found Kamar Aiken down the sideline and the wideout dived over the goalline for the score.
Tucker put the visitors ahead by seven by firing his 35-yarder through the posts, and the Browns were then dealt a blow when McCown was forced off with a shoulder injury.
© Getty Images
Cleveland were handed the chance to win the match when Schaub was picked off in the last minute by Tramon Williams, and Davis put his team just inside the range of Coons.
Coons lined up his kick, only to see his effort blocked by Brent Urban. The ball bounced into the path of Hill and the safety raced down the sideline and into the endzone to steal the win for his team as time expired.
The Ravens move to 4-7 in their season, while the Browns drop to 2-9.