DOUBLE-UP duo Robert Koren and Zoltan Gera fired Albion to within touching distance of the Premier League.
The Carrow Road encounter swung both ways but the Baggies held out admirably to stretch their unbeaten away run to eight games.
Albion roared out of the starting blocks and found themselves 1-0 up within 90 seconds as Koren clocked up his tenth goal of the season.
But Tony Mowbray's men failed to put the game to bed and were fortunate to see Jamie Cureton waste a hat-trick of glorious opportunities at the start of the second half.
Star-man Gera scored for the second successive game to double the lead with his tenth strike of the campaign in the 71st minute.
The visitors were forced to endure a nervy finish when Leon Barnett was harshly punished for handball in the penalty area and Ched Evans fired home from the spot.
But Mowbray's men admirably stood up to the challenge to bag a priceless win which means they need just one point from their final two games to clinch automatic promotion.
Roman Bednar was handed his first start in three games as he replaced the benched Ishmael Miller to partner Kevin Phillips in attack.
The Baggies made the perfect start and found themselves ahead within 90 seconds.
Phillips raced onto James Morrison's ball over the top and saw David Marshall fail to hold his low right-foot shot.
Both Bednar and City centre-half Jason Shackell challenged Marshall for the loose ball before it broke for Koren, who kept his cool to make room and fire just inside the right-hand post from eight yards.
Albion pushed forward again in the fifth minute when Phillips dispossessed Mark Fotheringham in the middle of the park, allowing Gera to set off on a mazy run, which resulted in him just failing to direct his 12-yard shot goalwards.
It was all Albion and, two minutes later, Gera fed Phillips who in turn found Bednar but the on-loan striker shot straight at Marshall from 15 yards.
The Canaries' first threat came in the eighth minute when Darren Huckerby's cross-cum-shot needed tipping over from under his own crossbar by Dean Kiely.
Kieran Gibbs got on the end of Jon Ostemobor's back-post cross in the 16th minute but couldn't keep his header down and watched it sail over the bar.
Six minutes later, the Baggies broke out of defence and Bednar found Phillips in space on the left-hand edge of the penalty area.
The former England man cut inside but scuffed his shot, which clipped off Gary Doherty into the air for Morrison to try a speculative overhead kick, only to be ruled offside.
Phillips played the ball into Bednar's feet on the edge of the area in the 32nd minute and he turned Doherty before mishitting his shot across goal, where Phillips missed the ball and was relieved to see the flag go up for offside.
Albion still looked bright in possession and Koren's pass found Bednar in the box six minutes later.
The Czech striker fired in a powerful drive which Marshall beat out and Shackell cleared.
Albion were perhaps unfortunate not to be awarded a penalty in the 41st minute when Doherty handled Phillips' pass.
But referee Iain Williamson deemed the offence to have taken place inches outside the box.
Phillips fired the resultant free-kick under the Canaries wall and Marshall had to dive to his right to stop the ball on the line.
The visitors made work for themselves in the first minute of added time when Kiely overcooked his throw out to Gera and Mo Camara pounced on the ball.
The City left-back fed it through to Cureton who found Gibbs and his back-post cross was fired wide of the right-hand post by Huckerby with the goal at his mercy.
Paul Robinson's last-gasp challenge may have distracted the Norwich striker but the Baggies defender suffered a cut head for his trouble and missed the remainder of the half as he went off for treatment.
Robinson returned after the break with a bandaged head and he immediately had work to do.
Huckerby got the better of him down the right and cut the ball back for Cureton, whose shot was blocked by Barnett.
A long punt by Marshall caught the Baggies' defence dozing in the 48th minute, with the ball bouncing all the way through to Cureton, who beat the stranded Kiely with a volleyed-lob which dipped inches over the bar.
Play immediately switched to the other end and Bednar managed to head Morrison's ball through to Phillips.
The Albion hitman looked sure to score before Doherty's last-ditch lunge poked the ball back to Marshall.
The Baggies were desperately unlucky not to go two-up in the 50th minute when Gera shook off his man and let fly with a 20-yard left-footed effort which thundered back off the bar and hit Marshall on the legs.
The City keeper then scrambled to his feet to gather before Morrison could poke home.
But Norwich were not out of this yet and, two minutes later, Evans headed straight at Kiely from Gibbs' left-wing centre.
In the 55th minute, Kiely was worked even harder when he dived low to his right to hold Cureton's header from Ostemobor's right-wing cross.
Cureton then chased down Evans' flick-on and looked sure to equalise - but smashed his 15-yard volley high and wide.
Glenn Roeder made a change in the 63rd minute as Dion Dublin replaced Cureton.
Mowbray responded with a double switch three minutes later as Miller and Luke Moore came on for Bednar and Phillips respectively.
That was quickly followed by another change as the weary Robinson made way for Martin Albrechtsen, who was returning from a two-game absence with a groin injury.
Albrechtsen was immediately in the action.
His diagonal free-kick in the 71st minute was headed down by Shackell and Gera pounced to volley a magnificent second into the bottom left-hand corner of the net from 20 yards.
But Albion's two-goal cushion was short-lived.
Just two minutes later, the referee's assistant flagged for an alleged handball by Barnett as he challenged Dublin in the box.
Berkshire official Williamson had waved play on but, after consulting his assistant, pointed to the penalty spot.
TV replays showed Barnett could hardly have got his arm out of the way but that was of no concern to Evans who fired home from 12 yards.
In the 78th minute, Shackell climbed highest at the back post from Ostemobor's cross but headed wide.
Albion finally looked to have wrapped the points up in the 90th minute when Miller finished off Morrison's low cross - but the giant striker was judged to be offside.
The Baggies then bravely endured five nerve-jangling minutes of stoppage time to make it three successive away wins.
NORWICH (4-4-2): Marshall; Ostemobor, Doherty, Shackell, Camara (Russell 84); Gibbs (Chadwick 75), Fotheringham, Pattison, Huckerby; Cureton (Dublin 63), Evans. Subs not used: Velasco, Pearce.
ALBION (4-4-2): Kiely; Hoefkens, Barnett, Clement, Robinson (Albrechtsen 69); Gera, Koren, Greening, Morrison; Phillips (Moore 66), Bednar (Miller 66). Subs not used: Danek (gk), Brunt.
GOALS: NORWICH - Evans (pen 73). ALBION - Koren (2), Gera (71).
BOOKINGS: NORWICH - Doherty (foul 76), Shackell (foul 81), Dublin (dissent 87). ALBION - Robinson (foul 44).
REFEREE: I Williamson (Berkshire).
ATTENDANCE: 25,442.
ALBION STARMAN: Zoltan Gera.
PIC: TWO GOOD...Gera celebrates his Carrow Road wonder strike with Koren and Miller
