Blues trailed 2-0 at the interval with Thomas Meunier and Joseph Akpala striking in a five-minute spell just before the interval.
But Jean Beausejour reduced the arrears and then won the penalty converted by substitute Marlon King which brought Blues level.
It means that Blues, Brugge and Braga are all level on seven points with two games remaining.
After 39 minutes Meunier took advantage of poor marking to put Brugge in front. No one picked him up as he attacked Victor Vazquez's corner and he had time to take the ball down on his thigh before driving the ball past Blues goalkeeper Colin Doyle.
Five minutes later Akpala doubled Brugge's lead after Blues were caught out by a quickfire counter-attack. He was left with a simple tap-in after Vazquez turned the ball across the six-yard box.
Nikola Zigic had a chance early in the second half to reduce the arrears but he failed to make a full contact on his attempted shot from a narrow angle.
The Serbian turned creator when teeing up a chance for Adam Rooney but his shot was blocked by Daan van Gijseghem.
Then after 54 minutes Beausejour gave Blues a lifeline with his first goal of the campaign. Zigic controlled a cross from Wade Elliott and his shot was blocked but Beausejour was alert to the rebound and fired past Vladan Kujovic.
In the 74th minute St Andrew's erupted when King levelled from the penalty spot after Ryan Donk was adjudged to have brought down Beausejour. Guirane N'Daw was booked before King sent Kujovic the wrong way from the spot.