Shane Long fired the Saints ahead after only four minutes, with Graziano Pelle also capitalising on some wayward defending to double the home side's lead on 38 minutes. Victor Wanyama knocked in a third 10 minutes after the break.
Andros Townsend scored a fine strike from 25 yards on 65 minutes, but it was little consolation after a miserable afternoon on the south coast for the travelling Toon Army.
Aston Villa slumped to the brink of relegation from the top-flight for the first time in 29 years following a 2-1 home defeat against Bournemouth.
Villa, who have won both the league title and FA Cup seven times in their history as well as the European Cup in 1982, have looked destined to drop out of the Barclays Premier League for weeks, sacking Remi Garde in March after five months in charge.
A 23rd league loss of a miserable league campaign was inflicted by a goal from Cherries defender Steve Cook in first-half stoppage time and an effort from Joshua King, who capitalised on more poor defending to score after 73 minutes.
Jordan Ayew netted a late consolation for Villa, firing in from 12 yards with five minutes left, but many home fans had already left the stadium.
Only Norwich's 1-0 defeat at Crystal Palace stopped their relegation from being confirmed, but Villa lie 15 points from safety with only 15 left to play for.
A Villa 1 - 2 B'mouth
So'ton 3 - 1 Newcastle
West Ham 3 - 3 Arsenal
Watford 1 - 1 Everton
Swansea 1 - 0 Chelsea
Man City 2 - 1 W Brom
C Palace 1 - 0 Norwich
Palace won for the first time in 2016 with a second-half goal from Jason Puncheon, which could prove enough to secure the Eagles' top-flight status.
Norwich, who had picked up successive victories to pull out of the relegation zone, felt they should have had a first-half penalty after Matt Jarvis appeared to be barged over by Damien Delaney, but could do nothing to prevent Puncheon curling in a fine 20-yard effort on 68 minutes.
Andy Carroll scored a hat-trick as West Ham fought back from 2-0 down to draw 3-3 with Arsenal at Upton Park, dealing another blow to the Gunners' fading title hopes.
Arsenal were in control following first-half goals by Mesut Ozil (18) and Alexis Sanchez (35), but the Irons levelled following a quick-fire double by Carroll just before half-time.
Carroll put West Ham ahead for the first time on 52 minutes, before Laurent Koscielny secured a draw for Arsene Wenger's men, who are now 10 points behind Leicester, who play relegation-battlers Sunderland on Sunday.
1 Leicester 69
2 Tottenham 62
3 Arsenal 59
4 Man City 57
5 Man Utd 53
6 West Ham 52
7 So'ton 50
8 Stoke 47
9 Liverpool 45
10 Chelsea 44
11 B'mouth 41
12 Swansea 40
13 W Brom 40
14 Everton 39
15 Watford 38
16 C Palace 37
17 Norwich 31
18 S'land 27
19 Newcastle 25
20 A Villa 16
Samir Nasri celebrated his first start for almost seven months with the winning goal as Manchester City came from behind to beat West Brom 2-1.
The Frenchman, fit again after a lengthy lay-off following thigh surgery, pounced from close range as City interrupted their Champions League quarter-final double-header against Paris Saint-Germain to claim three crucial Barclays Premier League points.
Stephane Sessegnon had fired the Baggies into a sixth-minute lead with a stunning strike at the Etihad Stadium but the same player then conceded the penalty that allowed Sergio Aguero to equalise.
Swansea beat Chelsea 1-0 at the Liberty Stadium to reach 40 points.
The Swans went ahead on 26 minutes through a volley from Gylfi Sigurdsson, while Chelsea had a goal by Alexandre Pato ruled out for handball early in the second half.
FA Cup semi-finalists Watford and Everton drew 1-1 at Vicarage Road, where both goals came in stoppage time at the end of the first half.
Midfielder James McCarthy put the visitors in front, but the lead did not last long as defender Jose Holebas headed the Hornets level.