Azaz fired home after pouncing on a quickly taken throw-in during the first half at St Mary's.
But Southampton needed a superb display from Israeli goalkeeper Daniel Peretz to preserve their vital victory.
Peretz made several fine saves, including a sensational double-stop in the second half.
Southampton move above Wrexham into sixth place in the table, sitting ahead of the Welsh club on goal difference with eight games left.
Tonda Eckert's team, who host Premier League leaders Arsenal in the FA Cup quarter-finals, are now unbeaten in their last 13 games.
Norwich are nine points adrift of the top six as their promotion hopes hang by a thread, despite 11 wins in their last 14 matches.