Roberto Di Matteo's side have made a surprisingly big impact following promotion from the Championship last season and they are now in the top eight after Cameroon midfielder Somen Tchoyi and Odemwingie sealed victory at The Hawthorns.
Peter Lovenkrands got one back for Newcastle in stoppage time but it was too late to stop his team suffering their third defeat in five games.
Newcastle defender Sol Campbell was making his 500th Premier League appearance, but the former Arsenal and England star was unable to keep Albion at bay.
Albion defender Paul Scharner had his side's first chance after a Chris Brunt corner was flicked onto the far post but he volleyed over from a narrow angle.
Shola Ameobi threatened for Newcastle when he headed a Danny Guthrie corner over the crossbar.
But Tchoyi broke the deadlock in the 32nd minute with his first goal at the Hawthorns and his fourth of the campaign.
It was a goal of ruthless efficiency. Brunt found Tchoyi in space inside the box and he cut inside Jose Enrique before curling a left-footed drive past Krul from 15 yards out.
Tchoyi was only playing due to Youssouf Mulumbu's suspension for his red card in last weekend's 4-1 win at Everton.
He was in dominant form and forced Tim Krul to turn a fierce shot from a narrow angle for a corner after good play by Odemwingie.
Newcastle boss Chris Hughton made a half-time change as he sent on Lovenkrands for Ameobi in a bid to spark his side into life.
The switch briefly made a difference as Newcastle pushed forward.
Wayne Routledge delivered an inviting cross from the right and Steven Taylor's powerful header was parried away by Scott Carson.
Then Jonas Gutierrez jinked his way into the box and saw his deflected cross-shot evade Carson, but Marek Cech was on hand to clear close to the goal-line.
Odemwingie doubled Albion's lead with his fifth goal of the campaign after 71 minutes.
Guthrie slipped over on the half-way line and left Odemwingie with a clear run on goal and he kept his composure before slipping a left-footed shot wide of Krul.
With a minute remaining, Odemwingie struck again as he took Cech's pass and skipped round Krul before firing home.
Lovenkrands scored from close range in the 90th minute but it wasn't enough to prevent Newcastle's first league defeat against Albion for 26 years.