Fulham, who are still searching for their first away win of the season, had to play the entire second half with ten men after the sending-off of Luis Boa Morte in first-half stoppage time.
Nathan Ellington set the tone when his eighth minute free-kick was beaten away by Fulham's Welsh international goalkeeper Mark Crossley.
Apart from a hopeful header by Brian McBride and a run by Collins John, Fulham never threatened in the opening period.
Albion thought their pressure had paid off after 27 minutes when Neil Clement floated a free-kick into the area. Ellington nodded the ball down into the path of Kanu and the Nigeria international rammed his close-range shot across the line only to look away in disbelief when he was flagged offside.
Fulham's task had looked hard enough with 11 men, but it became even more of an uphill battle when Boa Morte received his marching orders.
The Portuguese forward, who had been booked for an earlier challenge on Junichi Inamoto, was dismissed by referee Steve Bennett after collecting a second caution for a late lunge on Steve Watson.
It ensured that Fulham were to play the remainder of the match on the back-foot - content to try and catch Albion on the break.
It was a dangerous game to play especially as Albion did build up a head of speed at the start of the second half when Fulham's central defenders Carlos Bocanegra and Alain Goma toiled manfully to keep the Craven Cottage side on level terms.
Fulham's back line was cut in half after 54 minutes when Kanu and Inamoto combined to tee up Jonathan Greening whose wicked low cross was punched clear by Crossley.
But, apart from that one moment of menace, Crossley was well-protected by his back-line as Albion grew increasingly frustrated - especially when Kanu went sprawling under a challenge from Bocanegra only to see his penalty appeals waved away by Bennett.
Albion sent on their own battering ram in the shape of Geoff Horsfield for the final 19 minutes but he made little impact.
Fulham finally tested goalkeeper Tomasz Kuszczak for the one and only time nine minutes before the end of the game when the Poland international dealt comfortably with a long-range drive from Papa Bouba Diop.
MAN OF THE MATCH: Kanu (West Brom) - The former Arsenal marksman looked the player most likely to fire Albion into life but even the silky skills of the Nigerian could not unlock the Fulham defence.