United dominated for much of the game but Giroud scored with the Gunners' only shot on target to get the visitors out of jail at the death.
Juan Mata had given the hosts a well-deserved lead midway through the second half, but once again Jose Mourinho was left to rue yet another draw at home - their third in a row.
Both teams made a fast start
The game started at a frantic pace with both teams playing at a quick tempo as they tried to get off to a perfect start.
The Gunners though had the first big chance of the game when a corner kick, somehow the ball found it's way through to Alexis Sanchez, who was unmarked in the six-yard box, but he put his header wide when it seemed easier to put it on the target.
Cech to the rescue for the visitors
Apart from that chance, the visitors really struggled to create anything else of note and if it wasn't for Peter Cech at the other end then the hosts would have gone in at half-time with at least one goal.
Firstly he denied Mata with a fantastic diving save down to his left before he kept out a lovely curling effort from Anthony Martial which was heading into the top corner of the net.
There was one big moment in the first half that sent Mourinho and the United players crazy as Nacho Monreal made contact with Antonio Valencia in the penalty box, only for referee Andre Marriner not to give the penalty.
Overall though, Mourinho would have been happy with what he saw from his team in the first half as they gave him the energy that he wanted to see from his team.
United's energy continued at the start of the second half without creating any chances as Arsenal really struggled to create anything at all.
Mata gave United a deserved lead
After a period of time with Arsenal having most of the posession, United sprung into life, and finally got the goal that their play deserved during the game.
It came from a beautiful pass from Paul Pogba which found Ander Herrera in space, who then played a lovely cut back to Mata at the edge of the box, the Spaniard then stroked the ball into the net with some sublime technique.
They almost doubled the lead a few minutes later when Wayne Rooney forced the Arsenal defence into a mistake, but no one could get on the end of a brilliant cross by the striker.
Giroud gets the Gunners out of jail
Just as it looked like United were going to get over the line with an important three points, Giroud scored an equaliser in the last minute of normal time.
The goal was created by Alex Oxlade-Chamberlain, who played a lovely floated cross into the box after beating Marcus Rashford, which the big striker powered home.