The Nigerian capitalised on a mix-up between Ritchie de Laet and rookie keeper Declan Rudd, in for the suspended John Ruddy, to fire the Baggies ahead on three minutes at Carrow Road.
Norwich, now without a victory in four league games, had a few half-chances but did not really test Ben Foster - and would have been out of contention with 14 minutes left but for a fine penalty save from Rudd.
West Brom had lost all three of their Premier League games before this afternoon - but two of those were against title hopefuls Manchester United and Chelsea, while Stoke stole a 1-0 win in the last minute at The Hawthorns before the international break.
It took the Baggies only three minutes to go ahead following a terrible mix-up in the Norwich defence.
A long ball from Nicky Shorey caught the Norwich rearguard cold, as De Laet's header did not reach a hesitant Rudd, and Odemwingie nipped in to poke the ball into the far corner.
Norwich looked for a response, with Andrew Surman getting in on the left and his near-post effort was kicked out by Ben Foster, with captain Grant Holt unable to guide the rebound on target.
Norwich continued to press, but without really stretching the West Brom defence, while looking vulnerable on the counter.
Surman drove in a low 20-yard effort, but Foster was down quickly to make a comfortable save.
Elliott Bennett slammed Marc Tierney's left-wing cross over from eight yards after 30 minutes.
The former Brighton winger did better when cutting inside to sting Foster's hands with a rasping drive from the edge of the penalty area.
Rudd lifted his confidence with a decent block from Odemwingie as the West Brom forward burst clear down the right after the Norwich defence was easily split once again.
West Brom were inches away from going 2-0 ahead five minutes before the break when Graham Dorrans strode forward and lashed a 25-yard effort against the post.
Norwich had more purpose at the start of the second half, and worked the ball up to Chris Martin in the right side of the West Brom box, but his cut-back was hooked over by Andrew Crofts.
Bradley Johnson let fly from 25 yards and his effort flew inches over the top-right corner.
Norwich manager Lambert - who had not named a keeper on the bench - made a change on the hour as Wes Hoolahan replaced the lively Surman.
The Canaries were now enjoying their best spell, but again opted for the wrong pass when well placed in the final third.
Lambert made a double change with 20 minutes left as captain Holt was replaced by Steve Morison, and James Vaughan came on for Martin to make his first appearance since a summer move from Everton.
On 75 minutes, West Brom swung a deep free-kick into the Norwich area, which dropped to Steven Reid, who went down under what looked minimal contact from Wales forward Morison.
Referee Mark Halsey, though, had seen enough and pointed to the spot.
Odemwingie rolled the penalty towards the bottom left corner - but Rudd dived full stretch to produce a magnificent save.
Norwich were lifted by the sense of justice served, and moved on the offensive again.
Wales striker Morison planted a free-header wide from six yards after Tierney whipped the ball in from the left when he really should have hit the target.
In stoppage time, Vaughan was flatted by an elbow from Gabriel Tamas at a corner, which was missed by the referee and just about summed up Norwich's afternoon.
The Canaries continued to push West Brom back in desperate search of an equaliser.
Reid bundled over Morison on the edge of the area - but Johnson's free-kick hit the wall.