Clay's grab from Ryan Tannehill with 2:53 remaining erased a four-point deficit for the Dolphins (6-6), who later added a 27-yard field goal from Caleb Sturgis amid snow and cold at Heinz Field.
But Miami barely emerged with the victory after a bizarre finish on the game's final play.
With Ben Roethlisberger (23-of-30 passing, 349 yards, three TDs) facing a fourth--and-9 from his own 21-yard line with the clock expiring, the signal-caller completed a pass along the right sidelines.
The Steelers (5-7) then began a series of laterals. After getting the ball back, Roethlisberger lateraled to Antonio Brown, who shed some tackles and ran into the Miami end zone for an apparent game-winning score.
Game officials, however, determined Brown had stepped out of bounds at the Miami 13-yard line, thus ending the game.