Odion Ighalo (46) put the Hornets ahead just after the restart, before defender Craig Cathcart (58) made it 2-0.
Although Jamaal Lascelles (71) reduced the deficit, there was no way back for the Magpies, who are 18th.
Charlie Austin netted with his first touch in a Southampton shirt as a 1-0 victory for Saints piled the pressure on beleaguered Manchester United manager Louis van Gaal.
United supporters once again vented their anger towards Van Gaal after watching their team put on yet another dull, uninspiring performance at Old Trafford.
Boos rang out around Old Trafford at half-time and following the final whistle as Saints recorded only their second win at United's home since 1988, but also their second in 13 months.
There was no doubting who the home supporters blamed for this latest insipid display. Van Gaal was jeered at as he walked down the tunnel.
Adam Lallana's last-minute strike gave Liverpool an astonishing first Premier League win of 2016 in a 5-4 success at Norwich.
Barclays Prem
Leicester 3 - 0 Stoke
C Palace 1 - 3 Tottenham
Norwich 4 - 5 Liverpool
Man Utd 0 - 1 So'ton
S'land 1 - 1 B'mouth
West Ham 2 - 2 Man City
W Brom 0 - 0 A Villa
Watford 2 - 1 Newcastle
Barclays Prem Results
Jurgen Klopp's men had trailed 3-1 with under 30 minutes to go, then led 4-3 before Sebastien Bassong's stoppage-time strike levelled matters.
But there was still time for substitute Lallana to send a shot into the ground and secure a famous victory as Liverpool's own defensive failings were eclipsed by Norwich's at Carrow Road.
Klopp celebrated on the pitch with a shirtless Lallana, but there is much work to do if the Reds are to advance to the Capital One Cup final at the expense of Stoke, visitors to Anfield for the semi-final second leg on Tuesday night.
It had looked like new signing Steven Naismith would be an instant hero for the Canaries after Dieumerci Mbokani cancelled out Roberto Firmino's opener with a brilliant back-heel.
Naismith scored his first Everton goal against Liverpool and, on his debut following his move from Goodison Park, the Scot struck again to give the Canaries a 2-1 half-time lead.
Poor defending allowed Norwich to further extend their advantage when Alberto Moreno felled Naismith in the box and Wes Hoolahan converted the resulting penalty.
Jordan Henderson reduced the deficit from a gaping hole in the Norwich defence and another appeared allowing Firmino to equalise with 27 minutes remaining.
James Milner converted for 4-3 after capitalising on Russell Martin's dreadful pass back before the late drama.
Leicester moved back to the top of the table - three points ahead of Arsenal, who host Chelsea on Sunday - after they beat Stoke 3-0 at the King Power Stadium.
The Foxes took the lead just before half-time through midfielder Danny Drinkwater (42).
Striker Jamie Vardy (66) netted his 16th goal of the season on the break, before Leo Ulloa (87) added a late third.
Sergio Aguero scored twice to rescue a point for stuttering Manchester City in a 2-2 draw at West Ham.
The title challengers twice fell behind to goals from Enner Valencia and needed Aguero to help them out with a penalty and a late equaliser.
Valencia fired the Hammers into the lead after just 55 seconds, and put them back in front early in the second half with City's defence struggling once again.
But Aguero came to the rescue to deny West Ham an unlikely double over City this season.
Midfielder Dele Alli (84) cracked in a superb late strike as Tottenham beat Crystal Palace 3-1 at Selhurst Park.
The Eagles went ahead through an own goal from Spurs defender Jan Vertonghen (30), but in-form Harry Kane levelled (63) with a close-range header.
Barclays Prem
1 Leicester 47
2 Man City 44
3 Arsenal 44
4 Tottenham 42
5 Man Utd 37
6 West Ham 36
7 Liverpool 34
8 So'ton 33
9 Stoke 33
10 Watford 32
11 C Palace 31
12 Everton 29
13 W Brom 28
14 Chelsea 25
15 B'mouth 25
16 Norwich 23
17 Swansea 22
18 Newcastle 21
19 S'land 19
20 A Villa 13
After Palace midfielder Mile Jedinak hit the crossbar, Alli secured all three points for the visitors when he crashed in a volley from the edge of the box. Nacer Chadli scored a third deep into stoppage time.
Relegation-battlers Sunderland took what could yet prove a valuable point with a 1-1 draw against Bournemouth at the Stadium of Light.
The Cherries went ahead through £10million striker Benik Afobe (13), but the hosts, in 19th place, levelled in first-half stoppage time via Dutch defender Patrick van Aanholt
Bottom club Aston Villa battled through a goalless draw at West Brom, but remain cast adrift.