Darwin Nunez had already struck the post with a magnificent curling drive by the time Curtis Jones nodded the hosts in front from full debutant James McConnell's cross.
Arsenal, looking to lead the table for the first time since December, went in front through Declan Rice's flicked header. The home side continued to dominate possession and Havertz skewed wide their best opportunity to go further ahead.
The home side ought to have taken the lead on seven minutes when Gautier Lloris, in front of goal with the keeper out of commission, placed his finish straight at Carlos Soler, covering on the line.
Cesar Azpilicueta headed the Blues 3-2 in front in the 87th minute before Michy Batshuayi sealed the victory in stoppage time despite an unconvincing performance from the champions.
However, the home side ended the game with 10 men as defender Jan Vertonghen was sent off for catching Dortmund substitute Mario Gotze with a swinging arm.
England captain Harry Kane, inevitably, was involved with a deft touch to release Kingsley Coman, who easily beat exposed United keeper Andre Onana in front of the Stretford End.
Deyverson, who also scored against Barcelona in September, exploited dreadful Madrid defending to give Alaves the early breakthrough, but he was culpable for the equaliser, conceding a penalty which Ronaldo thrashed home.
Makeshift left-back James Milner played a through ball into the space down the middle, which Daniel Sturridge opened up into the path of Firmino who calmly slotted home.
Jurgen Klopp's side have only been beaten once so far this season and went two points clear of Arsenal at the summit despite some wastefulness in front of goal.
With fit-again Michael Olise subbed on in the second half for his first appearance of the season and Eze electric, the momentum seemed to be firmly with the home side.
The Cherries went in front at the end of a scrappy first half when Justin Kluivert's shot squirmed under Fulham goalkeeper Bernd Leno after a brilliant mazy run from Alex Scott to create the chance.
A defeat in Istanbul against Galatasaray - a team they have never beaten away from home in the Champions League - on November 29 will see United exit at the group stage.
Eberechi Eze converted from Tyrick Mitchell's cut-back to finish off a flowing Palace move early on and the visitors were deservedly in front at the break.
Bournemouth created a plethora of chances with top scorer Dominic Solanke, Alex Scott, Kerkez and Justin Kluivert all testing home keeper Jose Sa, who performed admirably in the face of constant pressure.