Leeds moved to within two points of Newcastle after beating Exeter Chiefs 27-22 to give their survival chances a massive boost.
Fly-half Adrian Jarvis booted 17 points and Hendre Fourie and Alfie To'oala claimed tries in either half as the Yorkshiremen ground out a priceless win.
Neil Back's men face Newcastle at Kingston Park on Saturday in a potentially decisive encounter between the northern foes.
Exeter fly-half Gareth Steenson opened the scoring with his first penalty of the match in the 14th minute but Jarvis quickly claimed two of his own to put Leeds 6-3 ahead.
In the 32nd minute, Fourie crossed the line to cap a period of sustained pressure in the left corner for a try which Jarvis converted.
However, Steenson added another penalty goal from distance to reduce the deficit and Jarvis did likewise on the stroke of half-time to leave the scores finely poised at 16-6 at the break.
Shortly after the restart, Chiefs wing Nemani Nadolo went close but Bryan Rennie followed up to score and Steenson converted before adding a penalty in the 51st minute to level the scores.
Jarvis then added another penalty in the 56th minute to put Leeds back in front and To'oala then touched down among a thicket of players.
Jarvis failed to convert but added a drop goal eight minutes from time and although Exeter hit back with a Steenson drop goal, Leeds held on.
Tries: Fourie, To'oala Vaeluaga
Pens: Jarvis 4
Pens: Steenson 4
Leeds Carnegie: 15 Michael Stephenson, 14 Lee Blacket, 13 Henry Faafili, 12 Luther Burrell, 11 Peter Wackett, 10 Adrian Jarvis, 9 Scott Mathie, 8 Alfie Tooala, 7 Hendre Fourie, 6 Kearnan Myall, 5 Marco Wentzel, 4 Dan Browne, 3 Phil Swainston, 2 Steve Thompson, 1 Mike MacDonald.
Replacements: 16 Phil Nilsen, 17 Gareth Hardy, 18 Juan Gomez, 19 Danny Paul, 20 Rhys Oakley, 21 Dan White, 22 Christian Lewis-Pratt, 23 Semi Tadulala.
Exeter: 15 Luke Arscott, 14 Nemani Nadolo, 13 Jason Shoemark, 12 Bryan Rennie, 11 Sireli Naqelevuki, 10 Gareth Steenson, 9 Junior Poluleuligaga, 8 Richard Baxter, 7 James Scaysbrook, 6 James Phillips, 5 Chad Slade, 4 Tom Hayes (c), 3 Chris Budgen, 2 Neil Clark, 1 Brett Sturgess.
Replacements: 16 Chris Whitehead, 17 Ben Moon, 18 Hoani Tui, 19 Dave Gannon, 20 Tom Johnson, 21 Haydn Thomas, 22 Ignacio Mieres, 23 Phil Dollman.
Referee: Tim Wigglesworth