The victory has moved Arsenal into 10th position in the Premier League table, now just four points off fifth-placed United, who missed the chance to close the gap on Chelsea in fourth.
Pogba had been expected to return against the Gunners having been left out against Burnley, but the France international was not even in the squad, and Scott McTominay's continued absence meant that Matic kept his spot at the heart of the midfield alongside Fred.
Arsenal's XI, meanwhile, included Pierre-Emerick Aubameyang, Mesut Ozil, Alexandre Lacazette and Pepe, while Granit Xhaka was selected in the middle of the park despite the speculation surrounding a potential move to Hertha Berlin during the January transfer window.
There was a red-hot atmosphere inside the stadium as Arsenal got the clash underway; United had the first shot of the match inside 30 seconds, although Marcus Rashford's drive from distance was comfortably gathered by the waiting Bernd Leno.
The visitors managed to race clear moments later through Daniel James, but the Welshman, who had gathered a flick from Anthony Martial, was brought to the ground by Sead Kolasinac. The resulting free kick was then delivered into a dangerous area by Fred, forcing Leno to punch clear under pressure.
Arsenal were having chances to force the issue, though, and the hosts made the breakthrough in the eighth minute when Pepe turned a low Kolasinac cross into the back of the net from close range. The summer arrival did not connect as well as he might have but the run of the ball deceived David de Gea.
Aubameyang then fired a volley over the crossbar minutes later following more smart work from Pepe, who was enjoying himself against Luke Shaw down the Arsenal right.
United's threat on the counter-attack was there for all to see, but the visitors were finding it difficult to force the issue in the final third, with Arsenal's lead still standing as the match passed 25 minutes.
The hosts were unhappy that the referee was prepared to let a number of fouls from United go unpunished in terms of cards, particularly considering that Kolasinac was booked in just the third minute.
Lacazette had the chance to send Mikel Arteta's side two goals ahead in the 32nd minute when he danced into the United box before facing up to De Gea, but the striker poked wide from a narrow angle before Lucas Torreira just missed the post with a low strike moments later.
The Gunners came close once again in the 38th minute following a poor pass from De Gea, but Pepe's effort from distance struck the foot of the post as United survived another dangerous moment.
Shaw was finding it incredibly difficult down the left with the home side continuing to threaten in that area of the field, and Harry Maguire had to come across to clear a low Ainsley Maitland-Niles delivery five minutes before the break.
The Gunners scored from the resulting corner, though, with Sokratis smashing into the back of the net from close range after De Gea had palmed a Lacazette header into a dangerous position.
Solskjaer resisted the temptation to make changes at the break; it must have been in the manager's thinking, though, with the likes of Shaw and Jesse Lingard in particular struggling in the first 45 minutes.
Fred tested Leno from distance in the 55th minute with United attempting to find a route back into the match, but the home side continued to look threatening on the counter-attack.
Pereira had a golden chance to halve the deficit shortly after entering the field, but the attacker hit the side-netting from a narrow angle with Arsenal escaping a dangerous moment.
Rashford then curled over the home side's crossbar in the 64th minute with United offering more of a threat inside a stadium that was becoming slightly nervous due to the flow of pressure.
United had a shout for a penalty 20 minutes from time when Aaron Wan-Bissaka went down inside the opposition penalty box, but neither the referee nor VAR thought that a foul had occurred.
Leno then had to keep out a strike from Greenwood, who was lively down the right for the Red Devils.
Lacazette struck over the United crossbar in the 78th minute after smart work from Aubameyang, before De Gra had to get down to keep out a strike from Reiss Nelson moments later.
Juan Mata was introduced shortly after with Solskjaer throwing caution to the wind, but it just would not happen for the 20-time English champions as the clock continued to tick towards 90 minutes.
Solskjaer cut a frustrated figure on the touchline with his team struggling to pick that final pass against an Arsenal side who were so dangerous on the counter-attack.
In the end, though, there were no more goals for either side, with the Gunners picking up a huge three points courtesy of a 2-0 success in front of their own supporters.
Arsenal are next in action at home to Leeds United in the FA Cup on January 6, while United will take on Wolverhampton Wanderers in the same competition on January 4.
ARSENAL (4-2-3-1): Leno; Maitland-Niles, Luiz, Sokratis, Kolasinac (Saka 69'); Torreira, Xhaka; Pepe (Nelson 62'), Ozil, Aubameyang; Lacazette (Guendouzi 81')
MANCHESTER UNITED (4-2-3-1): De Gea; Wan-Bissaka, Lindelof, Maguire, Shaw; Fred, Matic (Mata 80'); James (Greenwood 58'), Lingard (Pereira 58'), Rashford, Martial