Following a goalless first half at Ahmad bin Ali Stadium, Iran twice struck the woodwork shortly after the restart before Wales goalkeeper Wayne Hennessey was dismissed in the 86th minute.
Then, just as the game appeared to be ending in a stalemate, Cheshmi and Rezaeian struck deep into added time to secure a crucial three points.
Iran will next take on the US on Tuesday when a win would almost certainly see them advance to the knockout rounds for the first time in its history.
Iran lost to England 6-2 in its opener but Team Melli dominated the match against Wales and won the game to stay its hope alive.