Carlos Corberan's men climbed up to eighth, two points and placed outside the top six, after goals from Conor Townsend and Jed Wallace hit back against the Canaries.
The visitors, whose own play-off hopes had faded in recent weeks with dire form, led through Josh Sargent but the hosts struck back on the stroke of half-time through Townsend's wonderful free-kick - his first Hawthorns goal.
And the winner came through Wallace's swept finish from a nice move 10 minutes after the restart. The points were seen out professionally and comfortably and there was even time for an emotional Jake Livermore cameo from the bench after confirmation of his summer exit.
Corberan and his troops took the acclaim of the fans inside the stadium in a lap of appreciation at full-time with the Spaniard to take his side to south Wales to tackle Swansea on Monday week. Sunderland, Albion and Blackburn - in seventh, eighth and ninth - came make a final-day push for the top six – though the latter have a game in hand against Luton on Monday. Norwich, down from the top flight last term, lie down in 12th.
Just a two point gap to sixth leaves fans currently wondering what might have been - or what might still be - but in the context of Albion lying rock-bottom after Corberan's first fixture, it is an impressive achievement of note.
As expected, Corberan introduced John Swift back into his starting line-up for Taylor Gardner-Hickman. The latter endured a difficult evening at Sheffield United, after his mistake led to the Blades' opener.
The head coach felt Gardner-Hickman's energy over Swift could help at Bramall Lane, but reverted to regular started Swift against the Canaries.
Visiting boss David Wagner, like Corberan a former Huddersfield head coach, made four changes to his out-of-sorts Canaries side, with Sam Byram, Christos Tzolis, Jacob Sorensen and Dimitris Giannoulis all called for starters orders.
First-choice centre-halves Grant Hanley and Ben Gibson, captain and vice-captain, are long-term injury absentees.
It was a subdued atmosphere as the rain heaved down at The Hawthorns half an hour before kick-off. Results elsewhere involving Millwall, Coventry and Sunderland did little to inspire already-consigned Baggies.
For the second successive home game the Liquidator rang around the stadium to boost the atmosphere and a longer, extended edition did wonders for the volume levels.
Pumped-up decibel levels appeared to benefit both sides and the scoreline could easily have read 3-3 after 15 minutes in quite possibly an example of two sides playing freely without pressure and expectation.
Tzolis drew a routine Alex Palmer save from distance but US international striker Sargent was the real guilty party. He turned Erik Pieters inside out before somehow slotting the wrong side of the post.
As what felt like a game of basketball broke out Brandon Thomas-Asante really should have take advantage and made it double-figures for goals for the season after being superbly played in on goal by Swift.
The striker took the chance in his path and was one-on-one with Angus Gunn, but the Canaries No.1 came out on top with a strong save.
Sargent got the better of Pieters down the other end again moments later but could only head wide from a tight angle. Chances kept coming as Wallace struck narrowly wide from distance and Swift headed tamely at Gunn.
Swift caught the eye early on in a very watchable contest. The openings dried up, but not for too long as Thomas-Asante sent a poor left-footed curler off target from outside the box after good combination play with Swift.
Albion, for once, looked like a set-piece threat as Swift's corner was met by the towering Semi Ajayi but his header was palmed away by Gunn.
Moments later Albion fans and players had heads in hands at how they hadn't taken the lead.
The impressive Ajayi won the ball and strode forward before sending Wallace racing to the byline. His low cross fed Jayson Molumby whose goalbound strike was well blocked by away skipper Andrew Obomamidele.
Karlan Grant could not get his touch right for the rebound before Wallace was fed again on the right touchline. His cross, this time at head height, was a gem but just inches from Molumby's forehead.
While Corberan's men seem to have to move mountains to score a goal, the visitors managed it with ease.
Townsend was caught on the ball inside his own half and Albion never recovered. Dimitris Giannoulis fed Sargent with a simple pass towards the right of the box and the American slotted past Palmer this time with Pieters unable to get across while Townsend failed to recover.
It was a classic example of Albion this season - but this time they wasted little time to hit back.
There was little hope as Grant and Thomas-Asante left a bouncing ball outside the Canaries box to each other. Referee Bobby Madley decided the former was tripped and a number of blue and white shirts eyed up the dead ball.
It favoured a left-footer and Townsend was elected - and he stepped up with aplomb.
With all the guile of Neil Clement, the left-back bent an exquisite strike into the far top corner for Albion's first goal directly from a free-kick for 952 days since Matheus Pereira's effort at Everton in September 2020.
Little had come to pass in the early stages of the second period before the home faithful were celebrating again 10 minutes after the restart.
And the second goal owed almost everything to Thomas-Asante's workmanlike, never-say-die attitude.
The frontman was prone on halfway, surrounded by two or three yellow shirts, but he somehow came away with the ball down the left. Albion had the overlap and the striker fed Townsend near the byline.
Townsend's low ball looked to pick out Swift, who got minimum contact on his flick and the ball dropped perfectly into the path of the impressive Wallace to sweep home an excellent finish leaving Gunn no chance.
Albion and their fans were enjoying themselves and The Hawthorns was almost treated to a genuine goal of the season contender with Thomas-Asante showing a glorious touch and speed of thought.
He turned his defender on the halfway line with a marvellous touch and with open space ahead of him and Gunn off his line, the striker had one quick look and lofted a strike goalward. It dropped just a couple of inches, with the backpeddling Gunn nervous and helpless.
Wagner turned to his firepower from the bench with four changes including forward pair Adam Idah and Teemu Pukki but Albion showed impressive game management in several parts of the final 20 minutes.
The Canaries, who were in awful form going into the clash, failed to threaten at all with Corberan's Baggies comfortable. Sounds of discontent rang out from the away end.
The points long felt secure at 2-1 and there was time for an emotional and sentimental touch as the head coach satisfied supporters by turning to club captain Livermore.
The 33-year-old yesterday penned an emotional farewell letter ahead of his exit this summer and was introduced deep into stoppage time to a rapturous reception after six-and-a-half years as a Baggie. The club captain's name echoed around the stadium as Corberan and his players eventually took the acclaim on a lap of appreciation as they took the play-off battle to the final weekend.
41 - Goal Norwich. The Canaries lead as Josh Sargent is played in by Dimitris Giannoulis after Conor Townsend lost possession and Sargent this time makes no mistake.
45 - GOAL Albion! Baggies level as Conor Townsend makes amends by sending a peach of a free-kick into the top corner for his first Hawthorns goal.
55 - GOAL Albion! Carlos Corberan's side completed their turnaround victory with a goal down to Brandon Thomas-Asante excellent. He recovers possession while on the deck under pressure and eventually finds Townsend on the overlap. His low cross is met by Jed Wallace at the back post and the winger swept home emphatically.
Albion (4-2-3-1): Palmer; Furlong, Ajayi, Pieters, Townsend; Yokuslu (Gardner-Hickman, 75), Molumby; Wallace (Livermore, 90+4), Swift (Bartley, 84), Grant (Albrighton, 75); Thomas-Asante.
Subs not used: Griffiths, Faal, Malcolm.
Norwich City (4-3-3): Gunn; Marquinhos (Rowe, 66), Omobamidele (c), Byram, Giannoulis; Sara, Sorensen (Hayden, 66), Gibbs (Pukki, 66); Aarons, Tzolis (Idah, 66), Sargent.
Subs not used: Krul, Nunez, Hills.
Attendance: 24,638 (1,721 Norwich fans)
Referee: Robert Madley