After the visitors had kicked off attacking the River End goal, it took less than two minutes for them to take the lead.
David Marshall saved a snapshot from Kevin Phillips, but with him clutching the ball and lying on the ground, Roman Bednar fell onto him causing the City keeper to release.
Robert Koren seized the opportunity and planted a right-foot shot into the unguarded goal from six yards.
The goal lifted the visitors, who went on the offensive putting the home side under sustained pressure for the next 20 minutes.
A close offside decision denied Darren Huckerby the equaliser, but after the early onslaught by the visitors Norwich were well and truly giving as good as they were getting.
In the 38th minute, Marshall made a good save to deny Bednar.
The referee could have gifted the Baggies another goal when he harshly penalised Gary Doherty for hand ball.
However, Marshall did well to save the resulting 20 yard free-kick by Phillips.
Two minutes into the second, half Jamie Cureton missed another chance to equalise when his lobbed shot landed on the roof of the net.
At the other end, Zoltan Gera went even closer when his shot came back off the bar and was grabbed by Marshall.
Dean Kiely saved Cureton's rather weak header, then Huckerby's right-wing cross flashed across the face of goal with no City player on hand to apply the finishing touch.
Cureton blazed over when clear of the West Brom defence.
It came to nothing but there was another cheer from the crowd when veteran Dion Dublin replaced Cureton after 63 minutes.
Two minutes later, West Brom brought on Ishmael Miller for Bednar and Luke Moore for Phillips.
Shortly afterwards, Norwich were two down when Gera beat Marshall low to his right with a crisp dipping right-foot shot from 20 yards after Jason Shackell's header fell at his feet.
It took the linesman to spot the handball by Leon Barnett which enabled Ched Evans to score from the penalty spot with his right-foot shot sending Kiely the wrong way as the ball went to his left.