The Championship side had trailed to Joseph Akpala's third-minute opener but, roared on by more than 5,400 travelling fans, they equalised through David Murphy's slick finish in the 26th minute.
Then, following more than six minutes of treatment for defender Pablo Ibanez after he was knocked unconscious in worrying fashion, Wood stole in at the near post to score a dramatic winner.
The win saw City draw level with their hosts at the top of the group and fuel hopes of reaching the knockout stage.
One of several changes made by Blues boss Chris Hughton saw back-up goalkeeper Colin Doyle left out after his error against NK Maribor last month, with Boaz Myhill between the posts instead. Striker Nikola Zigic made his first European start of the season up front.
Blues were clear underdogs against the group leaders in Bruges, but were backed by a vociferous travelling contingent of 5,400 with many more in the city but without tickets.
Brugge made the early running and stunned the away side less than three minutes into the game.
Nabil Dirar whipped in a low cross which Akpala was able to guide home delicately.
Birmingham winger Chris Burke won an 11th-minute corner which saw Keith Fahey blast over from Wade Elliott's set-piece, while at the other end Jannes Vansteenkiste headed into the side-netting.
In the 26th minute Birmingham equalised.
After surviving a spell of pressure which saw Myhill save Vadis Odjidja's shot, they forced their way up field and, when Elliott sent in the cross, Murphy was on hand to guide the ball home.
Brugge's Jonathan Blondel and City frontman Zigic were booked in separate incidents, while Murphy tested Colin Coosemans with a free-kick.
Adam Rooney and Chris Burke squandered decent chances and the former should certainly have done better when he prodded Zigic's knockdown just wide.
Birmingham midfielder Guirane N'daw sent a highly speculative shot from distance towards the corner flag as the second half got under way.
Dirar remained a potent threat on the right flank as he created a chance for Akpala, who could only stab wide.
Zigic should have put his side ahead in the 59th minute when Burke's cross found him lurking behind the defence, but the Serb somehow headed wide of the far post.
Brugge boss Adrie Koster made two changes in a bid to seize the initiative, sending on Jimmy De Jonghe and Thomas Meunier before referee Daniele Orsato booked Brugge's Niki Zimling for handball.
Hughton matched his opposite number's ambition by sending on strikers Marlon King and Wood for Zigic and Rooney in the 73rd minute.
They teamed up immediately as Wood's ambitious shot was deflected to a grateful Coosemans. Brugge skipper Ryan Donk was not far off scoring a near-post header in the 80th minute.
There were anxious moments in the 83rd minute when Pablo Ibanez took a heavy knock in an aerial challenge with Akpala and his team-mates gesticulated frantically to pitchside medical staff.
N'daw and Burke looked badly shaken as the Spaniard, who appeared to be unconscious, received treatment for more than six minutes before being carried from the pitch on a spinal board, replaced by Liam Ridgewell.
Elliott sent a shot wide deep into added time before Wood stunned the entire stadium with a winner in the 100th minute.
Strike partner King whipped in the ball from the right and the young Kiwi stylishly guided the ball into the top right corner at the near post.