Tyne-Wear derby again next season
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.
City's title in the bag after 4-0 win over Villa?
Manchester City struck four second-half goals against a spirited Aston Villa side to put one hand on the Barclays Premier League trophy with a 4-0 win at the Etihad Stadium.
Edin Dzeko settled any nerves by breaking the deadlock after 64 minutes and followed up with his fifth crucial goal in the last three games.
1 Man City 83
2 Liverpool 81
3 Chelsea 79
4 Arsenal 76
5 Everton 69
6 Tottenham 66
7 Man Utd 63
8 Southampton 55
9 Newcastle 49
10 Stoke City 47
11 C Palace 44
12 West Ham 40
13 Swansea 39
14 Sunderland 38
15 A Villa 38
16 Hull City 37
17 West Brom 36
18 Norwich 33
19 Fulham 31
20 Cardiff 30
Stevan Jovetic and Yaya Toure - netting City's 100th league goal of the season with a quite brilliant solo effort - added further strikes in the dying moments to send City two points clear at the top, with just one game to play, in some style.
With a superior goal difference to Liverpool, City effectively need only a draw from their final game against West Ham on Sunday to claim their second title in three seasons.
After Liverpool's capitulation to Crystal Palace on Monday, City had been expected to cruise home.
That they found the going tough on a night of heavy showers was credit to Villa side who, safe from relegation, supposedly had little to play for.
Dzeko's heroics, in the absence of top scorer Sergio Aguero, further underlined his return to form and confidence this season after recent goals against Crystal Palace and Everton in the run-in.