PAUL Scharner&nbsp;again returned to haunt Everton as his first-ever Albion goal set them on the way to ending a five-game winless run in style.
Five of the Austria international's 15 English top-flight goals have now come against the Toffees and his 16th-minute opener was the catalyst for the Baggies equalling their biggest Barclays Premier League win on the road.
Chris Brunt doubled the visitors' advantage ten minutes later with an exquisite free-kick on his return from injury before Tim Cahill gave the Blues a lifeline three minutes before half-time.
But outstanding saves from Scott Carson and the 59th-minute sending-off of Mikel Arteta swung the game in the Baggies' favour.
Super-sub Somen Tchoyi grabbed his third goal of the campaign in the 75th minute - an absolute cracker - to all-but seal a second away league victory of the season for Roberto Di Matteo's men.
And Youssouf Mulumbu got in on the act three minutes from time when he added a sublime fourth - his third this term.
Sadly, the influential midfielder's afternoon turned sour just a minute later. After collecting a yellow card for over-celebrating, he was harshly dismissed within seconds for a second bookable offence.
But that failed to take the shine off Albion's first win at Goodison Park for more than 31 years.
Di Matteo made two changes to his starting XI as Pablo Ibanez and Giles Barnes dropped to the bench. Scharner moved back to centre-half and Graham Dorrans and Brunt came into midfield.
The Toffees threatened first in the second minute with Sylvain Distin just inches away from connecting with Leighton Baines' whipped-in free-kick.
Five minutes later, Cahill picked out Victor Anichebe in the penalty area and he forced Carson into a flying save with a curler from 15 yards.
But it was the Baggies who pinched an early lead in the 16th minute when Scharner powered in on Brunt's left-wing corner to get above Phil Jagielka and head home from ten yards.
Five minutes later, Peter Odemwingie teed up Brunt on the edge of the box and the Northern Ireland international blasted a low strike just past the left-hand post.
Gonzalo Jara felled Yakubu as the Nigeria international looked to race onto Steven Pienaar's throughball in the 24th minute. From the resultant Arteta free-kick, Carson flung himself to his right to beat the ball out.
Just two minutes later, Cahill fouled Mulumbu at the other end.
And Brunt, who only returned to training on Thursday following a groin injury, made the hosts pay the ultimate price as he curled a superb 25-yard free-kick into the top right-hand corner of the net.
Carson was fortunate to get away with a poor kick in the 38th minute as his clearance struck Cahill and rebounded straight to Yakubu, who was offside in front of goal.
Albion looked comfortable but the Toffees got themselves back into the game three minutes before half-time&nbsp;when the predatory Cahill leapt highest to nod home a trademark goal from Baines' left-wing corner.
The Blues came within a whisker of levelling moments later as Baines' flag-kick from the other side just evaded John Heitinga at the back post.
After the break, a swift Baggies counter-attack saw Morrison find Odemwingie and the former Lokomotiv Moscow striker cut inside Distin before firing over from 18 yards.
David Moyes made a double change in the 55th minute as Heitinga and Yakubu made way for Jermaine Beckford and Louis Saha.
A minute later, Beckford was slotted in on goal by Pienaar and the former Leeds striker saw his goalbound effort clip off the on-rushing Carson before being cleared off the line by the recovering Tamas.
That proved a pivotal moment because, seconds later, Arteta was adjudged to have stamped on Jara after the Chile international had disposed the Spaniard with a sliding challenge. Arteta was given his marching orders by referee Lee Mason following a lengthy consultation with his assistant.
Moyes made another switch on 61 as Jack Rodwell came on for Anichebe and Di Matteo followed suit five minutes later by bringing on Tchoyi for Thomas.
In between, Pienaar again threaded Beckford in on goal to prompt another fine save from Carson, who spread himself at the striker's feet.
The Merseysiders should have levelled in the 72nd minute when Beckford volleyed Hibbert's cross wildly over from point-blank range, when it seemed easier to hit the target.
But the Baggies deserved that moment of fortune and made the most of it.
They restored their two-goal advantage on 75 when substitute Steven Reid drilled a terrific ball out to Tchoyi on the left flank.
The Cameroon international drifted past Hibbert and Cahill before curling a delightful 18-yard strike into the top right-hand corner of the net.
There was to be one final scare for the visitors when Marek Cech missed his clearance from Saha's header and Beckford again&nbsp;failed to hit&nbsp;the target with a close-range overhead kick.
But three minutes from time, a memorable afternoon was complete when Tchoyi found Mulumbu in the box. The former PSG midfielder played a neat one-two with Morrison before coolly dinking home over Howard with the aid of a touch off Distin's chest.
Mulumbu's afternoon finished on a sour note a minute later when he was sent-off for a second bookable offence despite appearing to take the ball as he challenged Beckford.
But Albion played out the final&nbsp;minutes with little trouble - to bring some winter warmth to their 1,700-strong away following on a freezing afternoon.
EVERTON (4-4-1-1): Howard; Hibbert, Jagielka, Distin, Baines; Anichebe (Rodwell 61), Heitinga (Beckford 55), Arteta, Pienaar; Cahill; Yakubu (Saha 55). Subs not used: Mucha (gk), Bilyaletdinov, Coleman, Baxter.
ALBION (4-2-3-1): Carson; Jara, Scharner, Tamas, Cech; Mulumbu, Dorrans (Reid 73); Brunt (Barnes 90), Morrison, Thomas (Tchoyi 66); Odemwingie. Subs not used: Myhill (gk), Ibanez, Shorey, Fortuné.
GOALS: EVERTON - Cahill (42). ALBION - Scharner (16), Brunt (26), Tchoyi (75), Mulumbu (87).
BOOKINGS: EVERTON - Beckford (dissent 83). ALBION - Brunt (kicking ball away 31), Dorrans (foul 71), Mulumbu (over-celebrating 87).
RED CARDS: EVERTON - Arteta (violent conduct 59). ALBION - Mulumbu (foul, second bookable offence 88).
REFEREE: L Mason (Lancashire).
ALBION STARMAN: Scott Carson.
PIC: POINTING THE WAY...Scharner celebrates his Goodison opener with Dorrans&nbsp;
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