|Romelu Lukaku (left) scored the only goal of the game at a subdued Old Trafford|
Romelu Lukaku’s ninth league goal of the season, a 25th-minute header at the back post from a Juan Mata cross, was enough to give second-placed United the win but they had keeper David De Gea to thank for the three points.
The Spanish keeper kept out efforts from Charlie Williams, Dan Gosling and Ryan Fraser and blocked a fierce strike from substitute Jermaine Defoe as Bournemouth pressed for an equaliser that would not have been undeserved.
United, who rarely found much fluency at on a rainy and cold night, remained 11 points behind leaders Manchester City who won 4-0 at Swansea.
Elsewhere, Liverpool suffered more frustration at Anfield as they again dropped crucial home points after being held to a goalless draw by a resurgent and resilient West Bromwich Albion.
After receiving criticism for not starting with his key man Philippe Coutinho in the Merseyside derby draw against Everton, Liverpool manager Jurgen Klopp brought back the brilliant Brazilian but his side were still misfiring.
The Reds huffed and puffed without even managing a shot on target in the first half with Albion, producing a fine performance under new manager Alan Pardew, coming closest when Hal Robson-Kanu’s shot struck the top of the bar.
The home side, pushing hard for victory late in the second half, thought they had broken through only for Dominic Solanke’s bundled goal to be ruled out for handball.
West Ham meanwhile held out to secure a goalless draw at home with Arsenal after the visitors missed a string of chances in a pulsating Premier League clash.
West Ham’s brightest moment came in the 90th minute when substitute Javier Hernandez’s shot cannoned off the underside of the bar, against the run of play following waves of relentless Arsenal attacks.
Arsenal forward Alex Iwobi hit the post in the first half and midfielder Jack Wilshere blasted over the bar from eight metres after the break as the visitors piled on the pressure and forced a flurry of desperate clearances by West Ham.
The result left Arsenal seventh on 30 points from 17 games while West Ham, who beat champions Chelsea at home 1-0 on Saturday, are 19th on 14 points, behind Crystal Palace and West Bromwich Albion on goal difference.
Serge Aurier’s first goal for Tottenham and a late effort by Son Heung-min sealed a 2-0 win for the hosts over Brighton and Hove Albion to restore their place in the top four of the Premier League.
Aurier’s 40th-minute cross completely deceived Brighton goalie Mat Ryan and the ball drifted into net to break the deadlock on a largely frustrating evening for the hosts who struggled to break down the visitors.
Tottenham lacked zip against a defensively-minded Brighton and there were some jitters in the home defence after the break.
Tomer Hemed had Brighton’s best chance for an equaliser with a snap-shot but South Korean Son wrapped up the points with a glancing header late on from Christian Eriksen’s free kick for his fourth goal in four games.
Claude Puel’s Leicester City triumphed 4-1 at Southampton, the club who sacked him at the end of last season.