In a closely contested first half, in which Wijnaldum headed the Reds in front after just eight minutes of play, it was the hosts who lodged the only shot on target.
While City managed to produce an improved showing after the break, they were unable to find a way past Reds keeper Simon Mignolet as the home side held firm to claim a vital three points and consign Pep Guardiola to a fourth defeat in the Premier League.
Manchester United were pushed all the way to the wire as they rallied from behind to record a 2-1 victory over a gritty Middlesbrough at Old Trafford.
After a goalless first half, in which United had dominated proceedings for large passages of play, the deadlock was eventually broken when Boro's Grant Leadbitter scored against the run of play.
And, with time running out, United eventually drew level through Anthony Martial, before Paul Pogba headed home the match-winner just one minute later.
Chelsea recorded a 13th straight victory in their last match of 2016 after recording a 4-2 win over Stoke City in a thrilling encounter at Stamford Bridge.
The hosts went into the half-time break with a slender advantage, skipper Gary Cahill netting the all-important goal.
Bruno Martins Indi levelled matters just one minute into the second half, only for Willian to fire Chelsea back in front in the 57th minute.
Peter Crouch again levelled the scores with a strike from close range, but it was the Blues who had the last say as William bagged his second of the game, before Diego Costa made sure of the result with their fourth in the 85th minute.
Southampton ended the game with 10 men as a stunning Hal Robson Kanu strike condemned the Saints to a 2-1 home defeat, while Leicester City ended their barren run in the English Premier League with a hard-fought 1-0 win over West Ham United at the King Power Stadium.
Andre Gray netted a hat-trick in the space of 22 minutes as Burnley cruised to a 4-1 victory over Sunderland at Turf Moor and Swansea were condemned to their fourth consecutive loss of the season as Bournemouth cruised to a comfortable 3-0 win at the Liberty Stadium.