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Although unable to match their goal-scoring exploits from the 8-0 win over Sunderland last weekend, Ronald Koeman's side secured three points thanks to a goal from Sadio Mane.
Graziano Pelle was denied a second goal by the crossbar but their stingy defence held on well and the points allowed them to leapfrog Manchester City - they are now sitting three points behind Chelsea who take on Manchester United at Old Trafford on Sunday.
City were beaten 2-1 by West Ham in the lunchtime kick-off, with Diafra Sakho continuing his fine scoring run as Sam Allardyce's side retained fourth place.
Morgan Amalfitano gave the Hammers a 21st minute lead and Sakho scored in his sixth consecutive game to add to their lead with 15 minutes to go.
David Silva responded two minutes later but West Ham's defence held off a series of attacks to secure the points and leave City in danger of dropping eight points behind Chelsea.
Arsenal gave manager Arsene Wenger three Premier League points as a 65th birthday present as they beat Sunderland 2-0 at the Stadium of Light,
Alexis Sanchez continued to pay back his summer transfer fee with the opening goal of the game on the half-hour, and he sealed victory with his second in injury time.
West Brom snatched a 2-2 draw at home to Crystal Palace with the in-form Saido Berahino scoring from the penalty spot in injury-time of a controversial game.
Palace led 2-0 thanks to Brede Hangeland's header and a penalty from Mile Jedinak, but they were under pressure for much of the second-half after Victor Anichebe's 51st minute goal for the Baggies.
It took until injury-time for Berahino to level the scores, the penalty given by referee Mark Clattenburg for a foul on Anichebe - Palace were also left to complain about not being given another spot-kick in the first-half.
And there were no goals in the final 3pm kick-off with Liverpool held to a frustrating 0-0 draw at home to Hull City.
Under-fire striker Mario Balotelli missed the best chance of the game - fluffing his lines from three yards out - with just about the final kick and they drop into seventh.
Wilfried Bony scored both goals in the evening kick-off to lead Swansea City to a 2-0 win over Leicester at the Liberty Stadium.
Victory for Garry Monk's side lifted them above Liverpool into sixth.