Second-placed United had hoped to close the gap on Manchester City with a victory at Stamford Bridge. But instead they now sit 12 points behind the Premier League leaders as their title hopes fade following a run of one win from four games.
Chelsea boss Thomas Tuchel is unbeaten in nine matches in all competitions since replacing the sacked Frank Lampard, but the stalemate was a blow to his team's top-four challenge.
The fifth-placed Blues are one point behind fourth-placed West Ham and four ahead of sixth-placed Liverpool, who visit Sheffield United in Sunday's late game.
Neither side could claim they deserved to win in a game of few clear-cut chances. Just like at Old Trafford back in October, nobody could break the deadlock.
The biggest talking point came in the first half when United's appeal for a penalty was rejected by referee Stuart Attwell after he consulted the pitchside monitor when Callum Hudson-Odoi appeared to handle in a challenge with Mason Greenwood.
Giroud then just failed to meet a teasing Hudson-Odoi cross at the other end with Chelsea's best moment of the first-half.
The Blues showed more attacking intent in the second period and could have scored in the 49th minute when Ben Chilwell crossed to Hakim Ziyech in the middle, but David de Gea made a superb stop from close range. Victor Lindelof's brilliant clearance then denied the German international a tap in from Reece James's inviting cross with seven minutes to play.
Even though Fred tested Mendy a couple of times at the other end, United seemed to be happier with the draw. In the end, that is what both teams had to settle for.
Standout matches today