The striker, signed from Burnley during the summer, scored from close range four minutes into the second half after Birmingham had failed to deal with Joey Barton's free-kick.
Rangers, who gave debuts to on-loan Tottenham midfielder Tom Carroll and recent signing Matt Phillips, remain second in the Championship table behind leaders Blackpool on goal difference.
Harry Redknapp's side struggled to create clear-cut chances but were still able to register a fourth consecutive league win and have now not conceded a goal in more than six hours of Championship football.
Birmingham, meanwhile, have lost four of their last six games - the club's worst start to a league season for 25 years.
But they can consider themselves unfortunate, having caused unbeaten Rangers plenty of problems.
They were unlucky not to open the scoring when Andrew Shinnie's 23rd-minute free-kick was headed against the post by Lee Novak.
And Shinnie then wasted a glorious chance to put his team in front following a mistake by Barton.
Tom Adeyemi dispossessed Barton just inside the Rangers half and found Novak, who picked out Shinnie only for the midfielder to shoot woefully high and wide from 12 yards out.
City thought they had gone ahead shortly after the interval when Jonathan Spector completed a nice move by netting from close range, but he was offside.
A defensive mishap at the other end then presented Rangers with the opening goal.
Barton's delivery found Richard Dunne at the far post and after Adeyemi inadvertently sliced the ball against the underside of his own crossbar, Austin had the simple task of tapping into an empty net.
Barton later had an effort saved and hit the post in injury time with a superbly struck free-kick.
Phillips - on for the injured Junior Hoilett - burst through the middle but misplaced a pass to fellow substitute Jermaine Jenas, who would have been through on goal.
But Rangers were generally uninspiring and had their hands full as Birmingham continued to threaten.
Novak, who caused the R's defence problems throughout, headed wide from Chris Burke's cross before Kyle Bartley also went close with a header.
And the Blues felt they should have been awarded a late penalty after Spector went down under a challenge from Dunne.
Instead, the American defender was booked for diving, summing up a hugely frustrating day for Lee Clark's men.