United, Juventus, Roma and Basel all went through with victories in their final group games, but Atletico Madrid, beaten finalists in 2014 and 2016 were eliminated.
Jose Mourinho's United side came from behind to beat CSKA Moscow 2-1 at Old Trafford as they reach the knockout rounds for the first time since 2013-14.
Only a heavy defeat could have eliminated United in Group A, but Romelu Lukaku and Marcus Rashford scored two minutes apart in the second half to seal victory at Old Trafford, where they are unbeaten in 40 games in all competitions now.
At first glance Dzagoev appeared offside, but replays showed he had been played on by Daley Blind, who had just slid off the pitch. The goal was credited to Vitinho.
CSKA drop into the Europa League, while Basel reached the last 16 for the third time this decade, the Swiss champions beating Benfica 2-0 in Lisbon with goals from Mohamed Elyounoussi and Dimitri Oberlin.
Last season's beaten finalists Juventus progressed from Group D thanks to their 2-0 win away to Olympiakos in Greece, Juan Cuadrado opening the scoring early on before Federico Bernardeschi completed the victory late on.
Lionel Messi was left on the bench at a sparsely populated Camp Nou as Paco Alcacer headed the hosts in front on the hour and Jeremy Mathieu then scored a late own goal against his old side. Sporting drop into the Europa League.
- Chelsea drop to second -
==========================
Roma knew they would go through from Group C with a win, which they got, although they could only edge out Qarabag of Azerbaijan 1-0 at the Stadio Olimpico with a Diego Perotti effort early in the second half.
Saul Niguez headed Atletico in front on 56 minutes at Stamford Bridge, but Eden Hazard's ball across goal was turned into his own net by Stefan Savic to restore parity.
Diego Simeone's Atletico, who have only been knocked out by Real Madrid in each of the last four years, reaching at least the quarter-finals every time, must now accept a spot in the Europa League. They won that trophy in 2010 and 2012.
=============================
Bayern and PSG had already secured qualification from Group B but met at the Allianz Arena with first place still up for grabs.
Having lost 3-0 in Paris in September, Bayern needed to win by four goals to leapfrog Neymar and co.
They gave themselves a chance, with Robert Lewandowski and Corentin Tolisso both heading home in the first half.
Kylian Mbappe got one back just after the break, leaving Bayern with too much to do, although Tolisso did score again to make it 3-1.
PSG will be the happier side at winning the group, although possible last-16 opponents for them are Chelsea, Juventus and Real Madrid.
Liverpool are among the sides hoping to secure a place in the last 16 on Wednesday, with more four more berths still up for grabs for next Monday's draw.
Disclaimer: No Business Standard Journalist was involved in creation of this content
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