Their worst record since 1977, with no light at the end of the tunnel.
Four games to go and Alan Shearer said on MOTD: "They need a point to be safe, but I don't know where they will get it from!"
QPR and Burnley are probably destined for the Championship, but that third spot is anyone's guess.
Ayoze Perez gave the Magpies the lead in the 20th minute but Nelson Oliveira equalised in first-half injury time, and then Swansea were in front thanks to Gylfi Sigurdsson four minutes into the second half.
Jack Cork (71) deepened the gloom for the home supporters at St James' Park - and even though the fit-again Siem de Jong (87) got one back, Swansea hung on.
Substitute Fernandinho provided the late winner as Manchester City recovered after losing a two-goal lead to prevail 3-2 at home to Aston Villa in the Barclays Premier League.
On a day when Leicester and Hull were the significant winners elsewhere in the thick of the relegation battle, three more points for City took them up to second - still nine points behind champions elect Chelsea.
Manuel Pellegrini's team have gone above both Manchester United and Arsenal, before Sunday's fixtures between City's neighbours and Everton and the Gunners and the table-toppers.
Sergio Aguero's early goal, in just the third minute of Saturday's late match, took him one above Tottenham's Harry Kane to be the league's outright top scorer with a handful of matches remaining.
An Aleksandar Kolarov free-kick doubled City's lead in the 66th minute, only for the visitors' midfielder Tom Cleverley to reduce the deficit almost immediately with a shot from outside the area.
Barclays Prem
1 Chelsea 76
2 Man City 67
3 Arsenal 66
4 Man Utd 65
5 Liverpool 58
6 Tottenham 58
7 So'ton 57
8 Swansea 50
9 Stoke 47
10 West Ham 44
11 C Palace 42
12 Everton 41
13 W Brom 37
14 Newcastle 35
15 A Villa 32
16 Hull City 31
17 Leicester 31
18 S'land 30
19 QPR 27
20 Burnley 26
Carlos Sanchez (85) then levelled for Villa, with a sidefoot volley - but Fernandinho claimed the points a minute from time.
Villa therefore return to the midlands empty-handed and are still not clear of danger at the bottom, after Leicester and Hull eked out important wins away to Burnley and Crystal Palace respectively.
The Tigers and the Foxes both earned some breathing space, while others failed to do likewise on Saturday in a clutch of matches with a major bearing on who will drop out of the top flight next month.
Dame N'Doye missed a first-half sitter but put Hull in front seven minutes into the second half, and for good measure it was he too who made it 2-0 in injury time at Selhurst Park.
Leicester looked briefly in trouble against their bottom-of-the-table hosts at Turf Moor, where Matt Taylor had the chance to give Burnley the lead from the penalty spot just before the hour mark.
Instead, he hit the post - and within seconds Nigel Pearson's men snatched a 1-0 win with a goal at the other end as Jamie Vardy appeared to get the final touch, after the ball deflected goalwards off the Clarets' Michael Duff.
Sunderland drop into the bottom three after a 1-1 draw at Stoke.
Dick Advocaat's visitors were in front after just 58 seconds, thanks to Connor Wickham but Charlie Adam (27) made it all square at the break, and that was how it stayed.
QPR are still in big trouble after a 0-0 draw at home to West Ham.
The hosts did not make it easy for themselves when Charlie Austin's first-half penalty was saved by Adrian, but their own goalkeeper Robert Green did well to stop an Aaron Cresswell shot soon after the break.
It was 0-0 too between West Brom and Liverpool, Tony Pulis' side therefore banking another point to stay clear of worry and the Reds still in fifth.
In the lunchtime kick-off at St Mary's, Tottenham twice came from a goal down to earn a point in a 2-2 draw against Southampton.
Mauricio Pochettino was indebted to equalisers from Erik Lamela (43) and then Nacer Chadli (70) to eke out a point in the scrap with his old club to stay on the fringes of European qualification.
Graziano Pelle scored in each half for the Saints, in the 29th and 65th minutes.