Rotherham boss Steve Evans, however, was left fuming on the sidelines after his side were denied a penalty in their narrow Sky Bet Championship defeat.
Blues had led through first-half goals by Robert Tesche and Clayton Donaldson, before Matt Derbyshire pulled one back for fourth-bottom Rotherham.
However, Rob Kiernan's clumsy handling of Derbyshire near the hour mark saw the former Blues forward go down in the box only for referee Dean Whitestone to wave away appeals for a spot-kick, leaving the Millers boss visibly enraged as the home side held out.
Andrew Shinnie was unfortunate to be flagged offside from Donaldson's early flick-on while winger Demarai Gray's ball into the box from the left was dangerous but fruitless.
David Cotterill forced a good low save from Emiliano Martinez with a drive from just outside the area in the 19th minute as Gary Rowett's side continued to press.
The deadlock was broken in the 27th minute with a well-worked but uncomplicated attack
Tesche found space just outside the box and after a one-two with Donaldson, side-footed coolly into the bottom-right corner to leave Martinez with no chance.
Shinnie headed Cotterill's excellent cross over in the 38th minute after Donaldson's slight touch beforehand slightly changed the flight of the ball but eventually the second did come two minutes before the break.
Cotterill hit a free-kick from distance with trademark power and accuracy and when Martinez palmed it back into play, rather than for a corner, Donaldson pounced with an easy diving header.
Donaldson then fired a decent effort over on the turn as Blues sensed blood before the half was over.
Rotherham gained a foothold less than three minutes into the second half, however
Slack defensive play left Blues goalkeeper Darren Randolph in bother and when he only palmed Lee Frecklington's shot to Derbyshire, the deficit was halved as the striker tapped home with ease.
Frecklington then had a chance saved as Randolph kept out a deflected effort, while Martinez reacted quickly as Donaldson broke through at the other end.
Colourful Millers boss Evans was incandescent with rage on the hour mark when his side were denied a penalty, with Derbyshire bearing down on goal only to go down when tangling with home defender Kiernan as the centre-half looked to atone for an error seconds earlier.
Derbyshire was not far off with a 76th-minute header, while Donaldson and substitute Lloyd Dyer both came agonisingly close at point-blank range and Tesche twice fired straight at Martinez as Blues missed numerous chances to make the win more comfortable.
Source : PA