The only goal of the match came in just the third minute but both sides had plenty of chances throughout as the Baggies moved up to 12th on the table whilst the Canaries dropped to 17th.
Paul Lambert handed a full Premier League debut to 20-year old goalkeeper Declan Rudd, in the absence of the suspended John Ruddy, and it was a nightmare start for the youngster when teammate Ritchie De Laet's poor header allowed Peter Odemwingie to nip and slot home with his left foot.
The long ball from Nicky Shorey should have been better dealt with by De Laet but his attempt to hit it back to Rudd ended with Odemwingie, starting his first game of the season following an ankle injury, pouncing for the only goal of the contest.
Straight after the goal, Norwich were encouraging and could have leveled within five minutes when Andrew Surman was played in by Chris Martin only to see his cross knocked away by West Brom keeper Ben Foster and the deflection fell to skipper Grant Holt but it arrived too quickly and he could only put it wide.
The Canaries' next best chance came in the 34th minute when Elliott Bennett forced a decent save from Foster with a dipping ball from about 20 yards before the Baggies came close when Odemwingie found space but Rudd came up with a vital save with his legs just as it looked like the Nigerian would add a second to his tally.
It all opened up for West Brom's Graham Dorrans in the 40th minute and he produced a fantastic shot that had Rudd stranded but couldn't beat the far upright which came to Norwich's rescue before the Scot again made life difficult for the young keeper when he had another attempt which the keeper flapped at before it was eventually cleared away.
The other 'big' moment in the match came with 15 minutes remaining when West Brom were awarded a debatable penalty when substitute Steve Morison made minimal contact with Steven Reid in the area but Rudd came up with a huge save diving to his right to deny Odemwingie from the spot.
Norwich were getting desperate at the end and had a great chance when Morison's late header from another dangerous Marc Tierney cross from the left was inches wide but the home side were unable to equalise and left Roy Hodgson's men to pick up their first points of the campaign.
TRIBAL ASKS: Are Premier League referees appearing to award more 'soft' penalties this season? Are they too inconsistent with their calls?
Match facts: Second PL meeting at Carrow Road between two clubs after Norwich's 3-2 win in 2005.
Norwich have not beaten West Brom since November 2006.
West Brom have now kept 2 clean sheets in their last 40 PL matches but have scored in 15 of last 16.
Odemwingie's penalty miss was ninth spot-kick not converted of 15 awarded so far this season. There have now been four penalties in four matches awarded against Norwich.