The midfielder, making his Magpies debut after joining from Swansea, was involved in the build-up for both goals as Ayoze Perez and Georginio Wijnaldum put the hosts 2-0 up inside the opening 15 minutes.
Substitute Nikica Jelavic pulled one back for the Hammers shortly after the interval, but Newcastle held out and rose to 17th, two points clear of the drop zone as Swansea sunk to 18th.
Leicester returned to the top of the Barclays Premier League despite being held to a 1-1 draw at Aston Villa.
Rudy Gestede rescued a point for the rock-bottom hosts and denied the Foxes the chance to move further clear at the summit.
Shinji Okazaki opened the scoring but Mark Bunn saved Riyad Mahrez's first-half penalty and the Foxes were made to pay.
Arsenal will overtake Claudio Ranieri's side on goal difference with a draw at Stoke on Sunday with the Foxes a point ahead of the Gunners and Manchester City in their improbable title challenge.
The draw left battling Villa nine points from safety but boss Remi Garde can at least highlight improved performances in what seems like a futile struggle against the drop.
Sergio Aguero's brace helped Manchester City thrash Crystal Palace 4-0 earlier on Saturday at the Etihad Stadium.
Barclays Prem
Chelsea 3 - 3 Everton
B'mouth 3 - 0 Norwich
Tottenham 4 - 1 S'land
Man City 4 - 0 C Palace
Newcastle 2 - 1 West Ham
A Villa 1 - 1 Leicester
So'ton 3 - 0 W Brom
Fabian Delph opened the scoring with a shot in the 22nd minute that Palace goalkeeper Wayne Hennessey should have done better with.
Aguero then took his tally for the season to 13 goals with a deflected effort just prior to half-time. He finished a fine team move in the 68th minute, before teeing up David Silva to wrap things up six minutes from time.
Struggling champions Chelsea salvaged a point in dramatic circumstances from a thrilling clash with Everton as John Terry controversially made it 3-3 in the 98th minute at Stamford Bridge.
The home side were 2-0 down early in the second half thanks to a Terry own-goal and Kevin Mirallas' strike six minutes apart, but within 10 minutes of that they were level courtesy of quickfire efforts from Diego Costa and Cesc Fabregas.
Everton substitute Ramiro Funes Mori scored what looked set to be the winner in the 90th minute, but the stoppage time that then followed - so lengthy due to an injury sustained earlier by Bryan Oviedo - concluded with Terry's equaliser, over which there was a suspicion of offside.
Barclays Prem
1 Leicester 44
2 Man City 43
3 Arsenal 43
4 Tottenham 39
5 West Ham 35
6 Man Utd 34
7 Stoke 32
8 C Palace 31
9 Liverpool 31
10 So'ton 30
11 Everton 29
12 Watford 29
13 W Brom 27
14 Chelsea 25
15 B'mouth 24
16 Norwich 23
17 Newcastle 21
18 Swansea 19
19 S'land 18
20 A Villa 12
Bournemouth are now five points clear of the bottom three having seen off fellow strugglers Norwich 3-0 at home.
The Cherries traded places with the Canaries, moving up to 15th, after Dan Gosling's early header, Charlie Daniels' penalty and a first goal for the club for Benik Afobe sealed a comfortable victory.
Southampton are up to 10th following a 3-0 home win over West Brom, which was watched by new Saints signing Charlie Austin.
James Ward-Prowse notched a first-half brace with a fine free-kick and then a penalty before substitute Dusan Tadic put the matter beyond any doubt with just under 20 minutes left.
In the day's early kick-off, Tottenham consolidated their place in the top four as they came from behind to win 4-1 at White Hart Lane against Sunderland, whose new signing Jan Kirchhoff endured a debut to forget.
Spurs trailed to Patrick van Aanholt's 40th-minute effort but equalised within two minutes, Lee Cattermole making a hash of trying to keep Christian Eriksen's shot out on the line.
Defender Kirchhoff was introduced as a substitute in the 59th minute and saw Mousa Dembele give the hosts the lead within seconds, had Eriksen's shot loop in off him seven minutes later and then gave away a penalty with a tackle on Danny Rose which was converted by Harry Kane.