With just one league success against their name all season, the division's bottom club recruited the highly-rated McInnes from St Johnstone on Monday in a bid to turn around a dismal campaign that started under the management of ex-manager Keith Millen.
But McInnes found the same problems that faced his predecessor today, with his new side blunt in front of goal and poor at the back, with Burke's goals - one in the first half and the other with the last kick of the game - the decisive moments.
It meant another impressive turnaround for Chris Hughton's Birmingham, who won in Bruges in the Europa League on Thursday and now have a 100% record from their last five games - two of which have been on the continent.
They showed no signs of travel sickness at Ashton Gate, with Curtis Davies spurning a golden chance to put them ahead inside a minute, heading wide from a corner.
Nicky Maynard was off target from distance for the home side not long after, but Birmingham looked the more dangerous and Liam Fontaine did well to block a Burke shot, while David James held Keith Fahey's effort.
The former England goalkeeper may now be 41 but he quickly showed he has lost none of his assets, making a stunning double save to keep out Chris Wood - getting something behind two six-yard drives.
Bristol were allowing Birmingham to control possession and it was no surprise when the Blues went ahead nine minutes before the break, when Jean Beausejour crossed in from the right and Burke was able to beat James with a header.
Jon Stead scuffed wide from six yards as the Robins looked for a response before the break, while Neil Kilkenny tried his luck from distance after it.
Chances were few and far between in the second period, though, with Keith Fahey's drive for Birmingham - well blocked by Jamal Campbell-Ryce - the closest either side went, until James was again at his best to claw away a header from Nikola Zigic.
Albert Adomah then tried to make something happen for Bristol, but his injury-time strike was blocked clear by Stephen Carr and Birmingham duly went and added a second in the 95th minute, with Burke running on to a pass from Fahey and tucking away his fifth of the season.