Ajayi's gamewinner came one play after Miami quarterback Ryan Tannehill found a wide-open Jarvis Landry for a 32-yard gain, as the Dolphins (1-2) avoided what would be their first 0-3 start since 2011.
Tannehill threw for 319 yards and three touchdowns for Miami, and Landry caught seven of those passes for 120 yards.
Cody Parkey missed three field goals for Cleveland, including a 46-yarder as time expired in regulation. The Browns had a golden chance to win after Corey Lemonier stripped the ball from Tannehill and recovered the fumble himself with 20 seconds left.
But Parkey went wide left, some Dolphins fans who were leaving turned around in the stairwells, and to overtime they went.
Cleveland started Cody Kessler at quarterback, making him the third player in as many weeks to hold that job for the Browns -- and the fifth in their last five games going back to last season. The Browns lost Robert Griffin III and Josh McCown in the season's first two weeks, and ultimately got a spark from quarterback-turned-wide-receiver.
Terrelle Pryor became the first Browns player since 1977 to have multiple receptions, rushes and passes in the same game. The former Ohio State quarterback, now primarily a receiver, started taking some snaps in the game's second series and finished with eight catches for 144 yards -- along with 35 passing yards and 25 rushing yards. Pryor even played safety on the last play of the first half.
Miami trailed 13-10 at the half. Landry caught a 42-yard touchdown pass in the third quarter to put the Dolphins on top, punctuating that score by a leap into the goalpost and falling to the turf. And the lead got stretched to 24-13 when Tannehill hit Damien Williams with what became a 10-yard scoring throw early in the fourth.
Cleveland's first offensive touchdown in its last 16 series came with 10:12 left, when Pryor scored on a 3-yard run . The two-point conversion got the Browns within 24-21, and Parkey's 38-yard field goal later in the fourth pulled Cleveland into a tie.
Briean Boddy-Calhoun was inactive in Week 1 and Week 2 for Cleveland. He made his debut one to remember, becoming the first player in franchise history to run back an interception for a score in his first NFL game.
Cleveland won the coin toss prior to overtime and let Miami have the ball, even though a Dolphin touchdown on that possession would have ended the game. Miami wound up punting.
Dolphins C Mike Pouncey (hip) missed a third straight game, and his replacement -- Anthony Steen -- left with an ankle injury 15 seconds into the fourth quarter, so Miami turned to third-stringer Kraig Urbik. The Dolphins also lost LB Jelani Jenkins (quad), LB Koa Misi (neck) and TE Jordan Cameron (assessment for concussion).
For Cleveland, Kessler was shaken up briefly in the first half and removed by officials to be assessed in the fourth quarter after a big hit by Miami's Cameron Wake, who was offsides anyway. Kessler was checked out and cleared by doctors on the Browns sideline.