The home side carved out the best chances but United, who had a decent first-half penalty appeal turned down, defended solidly throughout to earn a point.
Chelsea remain in fifth spot with 44 points from 26 games, one point behind West Ham United, while Manchester United are second with 50 points from 26 games, 12 points behind runaway leaders Manchester City.
Liverpool ended a run of four successive Premier League defeats and scored their 7,000th top-flight goal during a 2-nil victory at Sheffield United that moves them within two points of a Champions League place.
Gareth Bale scored twice and made another as Tottenham cruised to an easy 4-0 win over Burnley that boosted their hopes of European qualification.
Arsenal also revived their European ambitions next season after a 3-1 win at Leicester City.
Arsenal are now 10th, while Leicester stay third.
Fulham and Crystal Palace drew 0-0.