The Nigerian struck in the 88th minute of an ill-tempered clash that saw Fulham end the game with nine men and Albion with ten, while referee Mike Dean also dished out six yellow cards.
Andy Cole fired Fulham ahead after 72 minutes - only to be sent off after a mass brawl five minutes from time. The punch-up was sparked by a foul from Albion midfielder Neil Clement on Cole, for which the Baggies man was ordered off.
The Cottagers striker aimed a blow at Clement and Dean quickly brandished his second red card in the space of a few seconds.
Fulham's Papa Bouba Diop became the first player to see red for pushing Albion central defender Darren Purse in the head after 60 minutes.
Before the late mayhem, Cole had looked set to end Fulham's run of three successive defeats with his fourth goal of the season.
Albion should have been out of sight before Cole struck - only to be denied by goalkeeper Edwin Van der Sar and a penalty miss by £3million new boy Robert Earnshaw.
Fulham were unfortunate not to break the deadlock after just three minutes when Albion's debutant goalkeeper Tomasz Kuszczak rather nervously pushed a header from Diop round the post.
While Albion were thankful to the intervention of Kuszczak, Fulham were even more indebted to Van der Sar, for ensuring they remained on level terms.
Twice in the space of 60 seconds, the giant Dutchman pulled off wonder saves to frustrate Earnshaw and Kanu.
In the 14th minute he tipped away a header from Earnshaw after a cross from Kanu, who was then denied by Van Der Sar when he powered in a header from Jason Koumas' corner.
The Cottagers should have taken full advantage of that double let-off when Collins John and Luis Boa Morte combined to set-up Cole, but the former Blackburn striker steered his shot wide from just three yards.
It should have been a miss that Albion punished when they were awarded a 29th minute penalty after Moritz Volz handled a header from.
Dean had no hesitation in pointing to the spot before booking John for dissent and Van der Sar for time wasting.
The long wait, and the fact that Van der Sar was some three yards off his line, appeared to get to Earnshaw who sent his shot high over the bar.
Fulham's luck then held again three minutes later when Kanu finally beat Van der Sar only to see his looping header crash to safety off the bar.
It was a slice of luck that the visitors made the most off, even after Diop was dismissed in the 60th minute for pushing Purse.
Despite their lack of manpower, Fulham looked a far more potent force in the second half.
Kuszczak pulled off a spectacular save from Ian Pearce in the 68th minute before Cole converted an Adam Green cross from close range.
It looked good enough to secure Fulham victory, but Albion kept battling and, after Clement and Cole had been dismissed, former Arsenal striker Kanu rose unmarked to deftly head home a corner from Koumas.
Man of the Match: Darren Purse proved to be a rock in the Albion defence as they battled hard to earn a point.