The Canaries took the lead 13 minutes into the second half through Dickson Etuhu but, after Sam Sodje's 73rd minute equaliser, a stoppage-time header from Diomansy Kamara netted a vital three points for the Baggies.
Norwich kicked off attacking the Barclay End goal and looked the stronger side in the early exchanges.
Lee Croft was booked for a foul on Paul Robinson in the 13th minute and seven minutes later Darren Carter's long-range shot came back off the bar.
This encouraged Robert Koren to try his luck but his shot dipped over.
Adam Drury was booked for a foul on Zoltan Gera after 23 minutes as the Baggies began to look the more likely to score.
Koran's shot grazed the bar, then the referee John Moss bizarrely waved play on when Croft was tripped just outside the visitors penalty area.
Youssef Safri limped off and was replaced by Mark Fotheringham four minutes before half-time.
Five minutes into the second half Jason Shackell became the third City player to go into the book.
Drury lost a tooth courtesy of Gera's arm but the event went unnoticed by Mr Moss.
Etuhu's near post header from Drury's left wing cross put the home side ahead and Darren Huckerby had a shot deflected for a corner before Robinson was booked for bringing down Chris Martin.
Gera brought a smart save from Tony Warner before Jason Koumas replaced Carter.
West Brom drew level when Fotheringham headed Sodje's goal-bound header off the line only for the same player to volley the ball into the roof of the net from some 12 yards.
Fotheringham had to make another goal-line clearance to keep out Jonathan Greening's header.
A body-check on Huckerby went unpunished but, as the game moved into stoppage time, substitute Richard Chaplow played the ball to Gera on the right of the City penalty area and his diagonal cross was met by Kamara's diving header on the six-yard line.
Substitute Dion Dublin then appeared to be flattened in the West Brom penalty area but Mr Moss turned away appeals for a spot kick.
Dublin did get in a goalbound header from Huckerby's right-wing cross but Dean Kiely pulled off an acrobatic save to secure a win.