Just weeks after admitting they needed divine intervention when they slipped seven points off the survival pace, the 46-year-old Uruguayan saw his team secure a fourth successive top-flight win for the first time since December 2000 to end their fears.
The win means 18th-placed Norwich have only a purely mathematical chance of avoiding relegation as the season draws to a close on Sunday.
Jack Colback's third goal of the season got the home side off to the perfect start when he turned Marcos Alonso's near-post cross past goalkeeper Ben Foster.
But Sunderland were cruising 18 minutes later when Fabio Borini volleyed home his ninth of the campaign to give the home side breathing space.
Neither goalscorer may remain at the Stadium of Light next season with Colback out of contract and being linked with derby rivals Newcastle, and Borini wanted back at Liverpool after a hugely successful loan spell.
But if their goals prove to be their last meaningful contributions - Poyet has hinted talks with Colback could yet be reopened - they could not have been any more timely.
The Stadium of Light was rocking as the bulk of a crowd of 45,181 celebrated the great escape as if it were the trophy the club came so close to securing in the Capital One Cup final at Wembley in March.
Sunderland could even afford a glaring miss from substitute Jozy Altidore, who steered a 65th-minute shot wide with the goal at his mercy as a chance to kill off a resurgent West Brom came and went.
But for all the visitors made a game of it in the second half after a tepid first 45 minutes, they rarely looked like denying their hosts three more points to go with the 10 they had garnered from their last four games against Manchester City, Chelsea, Cardiff and Manchester United.
Poyet's men had manoeuvred themselves into a position of strength, albeit a precarious one, by virtue of an unlikely sequence of results, and they ran out knowing a point would be enough to leave the Canaries needing to beat Arsenal handsomely on Sunday and hope the Black Cats lost heavily at home to Swansea to eradicate a goal difference deficit of 13.
However, having dealt in unexpected twists for much of the season, the head coach was not prepared to leave anything to chance and sent his players out under orders to seal their own fate.
They could hardly have responded to Poyet's pleas any more emphatically as they took the game by the scruff of the neck and raced into a potentially decisive lead with barely half an hour gone.
Colback needed just 13 minutes to set the ball rolling, turning up on cue to stab Marcos Alonso's low cross past Ben Foster after Sebastian Larsson had slipped away from Claudio Yacob in midfield and picked out the full-back wide on the left.
Confidence coursed through red and white veins as Sunderland, once again led admirably by midfield enforcer Lee Cattermole, dominated.
A surging break out of defence and slide-rule pass from the former Middlesbrough and Wigan man allowed Adam Johnson to deliver a dangerous 23rd-minute cross which Steven Reid had to hack away.
But as Reid and fellow full-back found themselves repeatedly under pressure, the Black Cats were relentless and extended their lead with 31 minutes gone.
Larsson was once again the instigator with a delicate lofted pass to pick out Borini, who volleyed first-time past the advancing Foster to make it 2-0.
The Baggies offered little as an attacking force in return, and it was not until injury time the home goalkeeper Vito Mannone was called upon to make a meaningful save when he pushed away Saido Berahino's drive.
West Brom boss Pepe Mel attempted to shore up his team at the break, replacing Reid with Craig Dawson and Yacob with Youssouf Mulumbu, and they immediately had a more compact and coherent look to them.
They came close to dragging themselves back into the game when striker Victor Anichebe only just failed to connect with Dawson's driven cross as he slid in.
Poyet replaced Johnson and lone striker Connor Wickham with Liam Bridcutt and Jozy Altidore with 61 minutes as he looked to consolidate, but with the visitors pressing, it was Black Cats old boy Stephane Sessegnon who dragged a shot wide two minutes later.
Altidore somehow missed the target when it seemed easier to convert Colback's 65th-minute cross and full-back Santiago Vergini smashed a shot across the face of goal four minutes later, but the points - and safety - were already secure.
Source : PA
Source: PA