Arsenal started the brightest with Nacho Monreal having a great opportunity early on but from there Louis van Gaal's side took over and Rashford was rewarded with a quick brace before Danny Welbeck gave the away side hope five minutes from the break.
It was a slow start to the second period from both sides before Ander Herrera fired home the third in the 65th minute which was followed by Mesut Ozil's effort four minutes later.
Oliver Giroud and Laurent Koscielny had half-chances towards the end but United held on for a crucial three points.
Arsene Wenger's side were quickly posed a threat as the match kicked-off and should have took the lead seven minutes in.
Ozil played a great first-time pass to the full-back but fluffed his lines with just David De Gea to beat as he thumped his effort straight at the Spaniard.
After that good chance, United began to dominate and Rashford marked his Premier League debut in style with a quick brace.
The first came just before the half-hour mark when Herrera played it out wide to Guillermo Varela whose cross was diverted by Gabriel but the 18-year-old striker showed great awareness to smash it beyond Petr Cech for his first league goal.
His and United's second came just three minutes later and showed the class of United's youngster's as the ball was brilliantly kept in by Varela after Juan Mata's mistimed pass.
Varela's header came to Jesse Lingard whose lofted ball was met by Rashford who headed it into the far corner with ease.
United looked to be cruising as the half came towards the end but the Gunners were handed a lifeline back into the match five minutes from the end.
It was poor defending on United's behalf as Ozil was allowed to float a simple free-kick into the area which was met by the unmarked Welbeck and despite De Gea's effort he couldn't keep out the former Devil's effort.
It was a slow start to the second period from both sides but Rashford was essential in United's third to restore their two-goal advantage.
The young striker was played down the right by Mata and did very well to hold up the ball before he laid it on a plate for Herrera just outside the area with a wicked deflection from the chest of Koscielny giving Cech no chance.
United looked to kill off the match and almost did in the 68th minute when another debutant Timothy Fosu-Mensah played it to Memphis who drove into the area before a quick one-two with Rashford but a touch from Hector Bellerin took it away from the Dutchman.
Wenger's side looked to make it a very nervy last twenty minutes when they got their second of the afternoon.
Alexis Sanchez's first real contribution of the afternoon was the excellent cross for Welbeck who looked to get his brace but De Gea managed to pull off a great save from his former teammate.
However he could only get it as far as Ozil who hit the follow-up into the ground which forced De Gea to watch as the German's effort bounced into the top corner.
Gunning for a equaliser
Arsneal had two half-chances as they desperatley searched for a equaliser to keep their title chances alive.
The frist came from Giroud as Sanchez flicked a dangerous ball to the back post but Giroud can't quite the right touch on it as headed it over the crossbar.
The second came just four minutes from end as a hoperful free-kick was floated into the area which was floated towards tyo Koscielny who looked to get it back across the face of goal and De Gea to hold on to collect a crucial three points.