The Group A leaders were all but guaranteed a place in the knockout stages before kickoff, although they could have still mathematically - if not realistically - missed out in a three-way playoff with Basel and CSKA.
CSKA gave their own hopes of qualifying a boost when they took a surprise lead in controversial fashion through Vitinho on the stroke of half time, but they were ultimately condemned to Europa League football after Christmas as quickfire goals from Romelu Lukaku and Marcus Rashford turned things around in the second half.
The result also sees the hosts rack up a club record-equalling 40th consecutive home game without defeat across all competitions, including 11 wins from 11 at Old Trafford so far this season.
United went into the match knowing that it would take defeat by seven goals - in addition to results elsewhere going against them - to miss out on a place in the last 16, and with such a strong position Jose Mourinho opted to make six changes ahead of Sunday's Manchester derby.
One of those saw Luke Shaw handed his first start of the season in any competition, and the full-back was among United's main attacking threats during the opening exchanges.
The first chance fell the way of Rashford after less than four minutes, though, and the 20-year-old should have done better when slipped through on goal by Lukaku only to see his scuffed finish bobble against the post.
Shaw continued to look like he had a point to prove to Mourinho and almost provided the opener in style when his first-time volley from Antonio Valencia's cross forced Akinfeev into a save, and the CSKA keeper then recovered in time to keep out Lukaku's follow-up effort from close range.
Moments later Shaw was on the charge down the left flank once again, but this time he got his final pass wrong as he put the ball too far in front of Rashford when the striker would have been left with a simple finish.
CSKA did manage to stem the tide against their wasteful hosts as the half wore on, and they even began to pose an attacking threat of their own as Vitinho and Vassili Berezutski both missed the target within five minutes of each other.
Indeed, the visitors took a shock lead right on the stroke of half time, although the goal came in controversial fashion as the final touch came off the back of an offside Alan Dzagoev following Vitinho's initial shot.
Confusion reigned after the ball had gone in as all eyes turned to the official, but the linesman appeared to rule that Daley Blind was playing Dzagoev onside despite having slid off the field moments before the final touch.
The situation then grew even more complicated as, having initially awarded the goal to Dzagoev, UEFA switched the scorer to Vitinho, who took the original shot after he had been found by Mario Fernandes's cutback.
Whatever the case, United went into the break behind despite controlling much of the half, but they came out quickly after restart and Rashford stung the palms of Akinfeev from a tight angle in the early stages of the second half.
Mata's hooked delivery was then sent glancing well wide by Chris Smalling before a powerful free kick from Rashford took a nick off the wall on its way behind, but United would not have to wait too much longer to turn the game on its head.
The equaliser came shortly after the hour mark when Lukaku registered just his second goal in his last 13 games for the club, skilfully steering Paul Pogba's lofted pass beyond Akinfeev with the outside of his boot.
Just 66 seconds later another goal drought came to an end as Rashford found the back of the net for the first time since October, drilling a low finish across goal and into the bottom corner after being found by Mata.
Mourinho immediately turned to the bench following the quick turnaround in a bid to rest players ahead of Sunday's derby, with Ander Herrera, Valencia and Lukaku all withdrawn in quick succession.
There was a chance for CSKA to rescue a draw from Old Trafford late on when Dzagoev was denied by the feet of Sergio Romero, but even that would not have been enough for them to reach the last 16 due to Basel's win away to Benfica in the group's other game.
United will now turn their attention to the weekend's huge derby against leaders Manchester City, while CSKA have a derby of their own against Spartak Moscow in the Russian Premier League - their final match before a three-month winter break.
Manchester United (3-4-3): Romero; Lindelof, Smalling, Blind; Valencia (tuanzebe, 72'), Herrera (McTominay, 66'), Pogba, Shaw; Mata, Lukaku (Martial, 74'), Rashford
CSKA Moscow (3-4-3): Akinfeev; Berezutski, Vasin, Ignashevich; Fernandes, Golovin, Kuchaev, Nababkin (Khosonov, 91'); Vitinho (Zhamaletdinov, 82'), Chalov (Gordyushenko, 66'), Dzagoev