Jose Mourinho's team have now gone four games without a win and remain in eighth place in the standings, while Arsenal are down to fifth despite having extended their unbeaten run in the top flight to 13 matches.
Arsenal were not getting shots away at the other end but the Gunners were having plenty of joy down the left flank as Sead Kolasinac and Alex Iwobi overpowered youngster Diogo Dalot, who was making his first Premier League start.
The Gunners had slowly begun to get on top and it came as little surprise when they took the lead, but their goal came as a result of a glaring error from David de Gea.
Mustafi got away from his marker to send a weak header towards goal and, although De Gea got both hands to the German's effort, the ball found its way over the line courtesy of the United keeper's fumble.
However, it took United just four minutes to find an equaliser. Leno made a solid save to keep out Marcos Rojo's free kick, but Ander Herrera retrieved the ball to cross for Martial, who lashed a shot into the net from six yards out.
Shots were few and far between for the remainder of the first half and after Rob Holding was forced off injured, five players picked up yellow cards for separate fouls in the space of 310 seconds.
United were the better side after the break but they were struggling to pose much of a threat to Leno, with only a 30-yard strike from Rojo forcing the German stopper into any action.
Arsenal slowly found their feet - just like they did in the first half - and as the match entered the final quarter, the visitors went back in front in equally calamitous fashion to their first goal.
Lacazette played a neat one-two with Henrikh Mkhitaryan before entering the penalty area and, although Rojo got across to make the tackle, the Argentine could only hit the ball against the forward and it subsequently rolled into the bottom corner.
However - like in the first half - United found an immediate equaliser. Arsenal failed to deal with a long ball from the kickoff and after Kolasinac diverted the ball towards his own goal, Lingard stole in to poke a shot past Leno.
Arsenal almost added a third goal in a matter of minutes, with Pierre-Emerick Aubameyang being superbly denied by De Gea from close range before Mkhitaryan somehow fired over from no more than three yards out.
The Gunners almost won it during the closing stages when Lucas Torreira was denied by De Gea, but a draw was a fair result as Arsenal extended their long unbeaten run and United were made to wait for another win in the top flight.
MANCHESTER UNITED (3-4-3): De Gea; Bailly, Smalling, Rojo (Fellaini 72'), Dalot, Herrera, Matic, Darmian; Lingard (Pogba 75'), Rashford, Martial (Lukaku 63')
ARSENAL (3-4-3): Leno; Sokratis, Mustafi, Holding (Lichtsteiner 36'); Bellerin, Torreira, Guendouzi, Kolasinac; Ramsey (Mkhitaryan 46'), Iwobi (Lacazette 65'), Aubameyang