The Nigerian capitalised on a mix-up between Ritchie de Laet and rookie goalkeeper Declan Rudd, in for the suspended John Ruddy, to fire the Baggies ahead on three minutes.
Norwich, now without a win in four league games, had a few half chances but did not really test Ben Foster, while Rudd made a fine second-half penalty save to deny Odemwingie a second.
Norwich manager Paul Lambert did not name a goalkeeper on the bench for the clash.
Youngster Declan Rudd deputised for the suspended John Ruddy, who must serve a one-match ban after his sending-off at Chelsea.
However, there was no place for academy keeper Remi Matthews, as Lambert instead packed the bench with forwards Steve Morison, Simeon Jackson and James Vaughan, back from a knee injury.
West Brom had striker Peter Odemwingie available after his recent ankle setback, and Shane Long also recovered from a calf problem.
However, there was no place for Zoltan Gera, available after a three-match ban, as the Baggies looked for a first league win.
West Brom may have lost all three of their Premier League games, 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 Ritchie 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 Grant Holt unable to guide the rebound on target.
After 16 minutes, referee Mark Halsey called back Paul Scharner to caution the West Brom midfielder after allowing play to continue following a late challenge on Bradley Johnson.
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.
West Brom were now enjoying more of the ball, as Youssouf Mulumbu drove a low shot just wide.
At the other end, 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.
Jerome Thomas then had a follow-up effort which Rudd spilled, and the keeper looked to have picked up a knee injury.
England Under-21 squad man Rudd was moving much more freely at the start of the second half.
Norwich 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.
There was a stoppage on 50 minutes when Dorrans needed treatment when Bennett caught him on the ankle in a 50-50 challenge, but the Scotland international was able to continue.
The Canaries pressed, but again lacked creativity around the West Brom penalty area.
Johnson then let fly from 25 yards and his effort flew inches over the top-right corner.
Lambert made a change on the hour as Wes Hoolahan replaced the lively Surman.
Norwich were now enjoying their best spell, but again opted for the wrong pass when well placed in the final third.
De Laet was left on the floor after being caught by Dorrans on the shin, with play held up for treatment to the on-loan Manchester United defender.
With 20 minutes left, West Brom sent on James Morrison for Dorrans.
Lambert made a double change with 20 minutes left as captain Grant Holt was replaced by Steve Morison, and Vaughan came on for Martin to make his first appearance since a summer move from Everton.
After 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 Steve Morison.
Referee Halsey, though, had seen enough and pointed to the spot.
Odemwingie smashed 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.
With seven minutes to go, Thomas limped off with cramp to be replaced by Chris Brunt.
Wales striker Morison planted a free-header wide from six yards after Tierney whipped the ball in from the left.
In stoppage time, Vaughan was floored by an elbow from Gabriel Tamas at a corner, which was missed by the referee and just about summed up Norwich's afternoon.
Reid bundled over Morison on the edge of the area, but Johnson's free-kick hit the wall.