Worcester centre Alex Grove got the only try at Sixways as they pitched Premiership relegation rivals Newcastle into even deeper relegation trouble.
In a dour Sixways struggle, Andy Goode kicked the Warriors' other 14 points, with the conversion and four penalties.
Jimmy Gopperth replied with three penalties for Newcastle, who remain eight points adrift at the bottom.
But Goode's final penalty in the last minute even denied the Falcons the consolation of a losing bonus point.
And, having led 9-6 just before the hour, their hopes of a first league away triumph since February vanished to edge them a little closer to the trap-door.
Eleventh-placed Wasps, who host Newcastle on the regular league season's final day in early May, are not expected to gain anything from their trip to play-off contenders Northampton on Saturday.
But Newcastle's latest defeat - their 13th from 18 Premiership starts this term - has upped the ante for their remaining fixtures at home to Sale and Saracens and away against Gloucester and Wasps.
Newcastle gave loan signing Peter Stringer his debut, the Ireland international scrum-half having arrived at Kingston Park after completing a similar short-term stint with league champions Saracens.
Worcester, meanwhile, welcomed back wing Marcel Garvey and number eight Kai Horstmann after injury lay-offs.
After a grim first half, enlivened only by a penalty for each side, referee JP Doyle's half-time whistle could not come soon enough for Worcester, who looked a pale shadow of the side that have beaten both Saracens and Gloucester at home this year.
Gopperth and Goode then swapped penalties again before Newcastle led for a third time on 54 minutes.
But Goode once again matched it after a break by Warriors wing Miles Benjamin, to leave the game tied up again entering the final quarter.
Worcester were now looking far more dangerous with ball in hand than they had been in the first half. And it came as no surprise when Grove crossed 15 minutes from time, his touchdown ending a prolonged spell around Newcastle's 22 that ultimately cracked open Falcons' well-organised defence.
Goode converted to stretch Worcester eight points clear of Wasps - and 16 in front of Newcastle, with just four games left.
He then slotted his fourth penalty just a minute from time, leaving the prospect of Newcastle playing Championship rugby under Dean Richards next season looking an increasingly likely one.
VIEW FROM THE DRESSING ROOM
Worcester coach Richard Hill:
"They gave us a tough old battle. I feel for them. Andy Goode's late penalty was a hammer-blow, but I still think there is a chink of light for them.
"We had to change tactics at half-time. We went into the match with a strategy of trying to play Newcastle at their own game, but that didn't work.
"We also needed to relax. I felt we were very edgy, but I hope the players can relax a little now after this win."
Newcastle rugby director Gary Gold:
"We are going to have to front up and be courageous now. We have four games to go and there is nowhere to hide. I knew what the task was when I came here, and I am completely up for the fight.
"A lot of aspects of our game were good, but it wasn't to be. I am unbelievably disappointed, absolutely gutted. It is devastating to leave here without even a bonus point."
"One try was the difference tonight. No defence can allow for missed tackles, and we missed one on our line for their try.
Worcester: Pennell; Garvey, Grove, Rasmussen, Benjamin; Goode, Perry; Mullan, Lutui, Taumoepeau, Percival, Chris Jones, Betty, Abbott, Horstmann.
Replacements: Cowan for Abbott (20), Arr for Perry (66), Shervington for Lutui (73).
Not Used: C Jones, Douglas, Gillies, Gray, Fatiaki.
Newcastle: Manning; Uys, Helleur, Fitzpatrick, A.Tait; Gopperth, Stringer; Golding, Thompson, Murray, Swinson, Goode, Tu'ifua, Welch, Hogg.
Replacements: Graham for Thompson (64), Shiells for Golding (66), Sheriffe for Uys (70), Fondse for Goode (70), Wilson for Tu'ifua (73), Goosen for Manning (75).
Not Used: Wells, Chudley.
Referee: Sean Davey (RFU).