Heavy rain fell throughout the game leaving the pitch difficult to manoeuvre and both sides struggling to string together a cohesive attacking move.
However, the story of the half was of penalties that might have been rather than goals. Chelsea called for a penalty on 40 minutes after Harry Maguire pulled down Cesar Azpilicueta in the box during a Chelsea free kick.
Replays showed Maguire with his arms wrapped around Azpilicueta from behind but neither the referee or VAR decided to award a penalty.
One minute later VAR was checking a potential foul on Rashford who had been clipped in the box by Thiago Silva. However, no foul play was deemed to have taken place and play continued.
United's new signing Edinson Cavani nearly made an explosive start to his first appearance for the side when he got on the end of a Bruno Fernandes corner as his first touch on 59 minutes. Fernandes drifted the ball into the box and Cavani directed it just skittering past the post.
Chelsea head coach Frank Lampard made a double substitution just after the 70-minute mark pulling big summer signings Timo Werner and Kai Havertz off after a quiet game for both and replacing them with Mason Mount and Tammy Abraham in the hope of shaking up the side's attack.
United were forced into some frantic defending on 85 minutes as the ball ricocheted around their box but Chelsea were unable to find a goal with Jorgino sending his shot over the bar.
Rashford had a stunning chance to win it for United in the dying moments but Mendy managed to just get a hand to his curling shot.
"I have the feeling we could have done more," Azpilicueta said after the game. "We found spaces but we didn't create the last pass or pick the right pass or make the right decisions. We defended solid and they have at the end one save by Edouard [Mendy] and in the first half one mistake by us, but we could do more.
"For moments we had spells. In the first half we controlled better but in the second half sometimes we rushed it and didn't move the ball good enough. We lost possession too easily which meant we lost some control but we showed solidity and it was a great effort from everybody. Obviously we want more."