Mexico, who have often struggled on their visits to the Caribbean, were beaten by a 14th minute Ricardo Fuller goal in Kingston.
"The boys stuck to the game plan and it worked well for us," said Jamaica's interim coach Theodore Whitmore, brought in after Rene Simoes was fired last month.
Mexico are second in Group Two with nine points from four games and are level on points with leaders Honduras.
Jamaica, despite their win, are a further five points behind.
YES RISE TO THE OCATION.......DI BOYZ DEM REALLY TEK IT TO DEM LAST NIGHT........MI LUV HOW DI TEAM PERFORM TRUS MI WE CUDA WIN BY BOUT 3-0 BUT U JUS SUH IT DID FI GUH.........AND WHO THE HELL IS JAMAL CAMPBEL-ROYCE DI YUTE REALLY TEK IT TO DEM LAST NIGHT.........HOW MI NEVA HEARD OF HIM.........WELL A 2010 MI YES..... WE A REPORTRIATE TO ZION