Azaz opened the scoring for Saints just six minutes into the game, receiving the ball centrally on the edge of the box from his teammate Leo Scienza.
The 25-year-old then shifted the ball before drilling a shot into the bottom right-hand corner of the net and past the reach of Blues goalkeeper James Beadle.
Azaz has now scored seven goals in the Championship this season, with six of those coming in his last five matches.
Moments later the home side went close again, Scienza dancing past defenders before pulling a pass back for Tom Fellows who crashed a powerful effort against the crossbar.
Southampton then made it two in the 24th minute as Azaz found Armstrong from outside the box and struck a right-footed shot across the keeper into the far left-hand corner of the goal.
Blues won a free-kick on the edge of the Southampton penalty box which was well struck by Tommy Doyle but Bazunu tipped the midfielder’s effort narrowly over the top of the crossbar.
Gray produced a stunning strike in the second half to fire Birmingham back into the game, cutting inside onto his right foot and then curling an effort into the far corner to make it 2-1.
But Saints were able to score again moments later as Scienza shot from the edge of the area and Armstrong followed up to tap home from close range as Beadle palmed it into his path.
Kyōgo Furuhashi had a great chance to cut the deficit with around 20 minutes remaining as Jack Robinson’s header rebounded off the post but his shot flew over the bar.