The Warriors had been pushed to the foot of the table after Leeds' victory over Gloucester on Saturday, but another battling display ensured their stay lasted a mere 24 hours.
The win came courtesy of tries from Giscard Pieters and Pat Sanderson plus nine points from the boot of fly-half James Brown, which included two drop goals of which Newcastle's Jonny Wilkinson would have been proud.
The England fly-half, who is set to be included in the British and Irish Lions touring squad on Monday alongside the returning Colin Charvis, was not involved at Kingston Park as he continues to recover from knee ligament damage.
Matthew Burke was left to provide the classy touches for Newcastle and his try with five minutes remaining earned Newcastle a bonus point, but they could not dampen the Warriors' passion to remain in the Premiership.
While Leicester must wait for the stage-managed drama of a Grand Final for the chance to seal a title they royally deserve, the tension at the other end of the Premiership is all too genuine.
Worcester, Leeds, Northampton, Harlequins and London Irish are all in the mire, embroiled in an almighty scrap against the drop. It is wonderful theatre and the punters have come flocking in.
And yet talk of ring-fencing the Premiership persists. Those in favour argue hard that it makes acute financial sense. Those against prefer the gut-wrenching drama.
Worcester jumped three places to join London Irish on 38 points and are one better than Harlequins and two ahead of Leeds and Northampton, who they play in the final match of the season. It is incredibly tight.
Worcester are desperate to avoid returning from whence they came and they played today as they have all season, with fight and spirit.
The club have long-term ambitions in this division, having signed Shane Drahm and Andy Gomarsall for next season, and they started with purpose in blustery conditions at Kingston Park.
Having spurned a penalty shot at goal, the Worcester forwards launched a heaving drive from the lineout and moved to within five metres of the line but winger Pieters spilled the ball with a try begging.
Charvis, returning to action after two months out with a broken foot, made quite an impression. His bullocking run down the middle got Newcastle in motion.
Hall Charlton was caught just metres short of the line but Brown was caught offside and Burke slotted the penalty to give Newcastle a 12th minute lead.
The game was scrappy and Worcester, as they have all season, were scrapping hard.
With half an hour gone Thomas Lombard broke through the middle and Pieters made amends for his earlier fumble as he held off the cover defence long enough to touch down in the corner. Swirling winds snagged Brown's attempted conversion from the right touchline.
Charvis was then involved in the second of two scraps inside five minutes. The first had been on the deck with Worcester captain Sanderson, the second a stand-up set-to with his opposite number Siaosi Vaili for which both men were sin-binned.
Given Charvis was the instigator referee Ashley Rowden was perhaps a touch harsh on the Samoan, doubly so given Burke landed a second penalty to regain the lead for Newcastle.
Burke then missed a chance from 40 metres out and Worcester snatched the lead as Brown landed a drop-goal three minutes into first-half injury time.
Mathew Tait, on his return to the side after a short holiday designed to freshen him after a tumultuous season, was replaced at half-time by Epi Taione but it was Worcester who made the first impact after the break.
Sanderson was halted by a high tackle from Luke Gross just two metres out and Brown slotted the penalty.
Newcastle hit back swiftly. A wonderful pass from captain Jamie Noon almost sent Tom May over. But the ball was recycled and Cory Harris burst through Dale Rasmussen's tackle to score in the left corner.
Burke failed with the tough touchline conversion and, with 31 minutes remaining, the match was intriguingly poised at 11-11.
Brown broke the deadlock with a sweetly struck drop-goal and Sanderson powered over as the Worcester lineout drive once again sent Newcastle sprawling.
Hands in the ruck had earned Worcester a kickable penalty, but they had enjoyed such dominance from the set-piece that Sanderson opted to go for touch.
Craig Gillies took a wonderful catch in the lineout and Sanderson touched down under a pile of bodies to open Worcester a 21-11 lead.
Burke scored in the corner to earn Newcastle a bonus point but Worcester took the win which keeps their Premiership dreams well and truly alive.
For Newcastle Falcons:Tries: Harris, Burke Pens: Burke 2
For Worcester Warriors:Tries: Sanderson, Pieters Con: Hayes Pen: Brown Drops: Brown 2