The Belgium midfielder, introduced at the start of the second half, fired Boro 2-1 ahead on 69 minutes before fellow substitute Malaury Martin put the game out of reach with a third.
Adam Rooney had given Blues the lead from the penalty spot, only for Barry Robson to level just after the restart.
Nathan Redmond had the first effort of the match but his shot from distance fizzed just past the post.
The home side started to come back into the game and it took a smart save from Boaz Myhill to prevent Robson giving the hosts the lead with a drive from 20 yards.
Myhill was continued to be tested as Boro started to dominate, this time producing a smart save from a deflected Emnes strike.
Against the run of play, Blues took the lead after 36 minutes from the penalty spot. Rooney fired into the bottom left corner after the lively Burke had been brought down in the box.
The home side equalised three minutes after the restart when Robson's free-kick crept in off the near post.
After replacing McDonald, Haroun made an instant impact and, after having a shot blocked, he netted the second goal by slotting in from 12 yards, before Martin drilled home from 25 yards.
Jean Beausejour almost grabbed a consolation in stoppage time but his effort from distance was easily saved by Carl Ikeme.