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Rangnick steps in to replace Solskjaer, who was removed after United's 4-1 defeat by Watford this month.
Chelsea claimed a thumping 4-0 victory over Juventus.
Former player Michael Carrick, a member of Solskjaer's coaching staff, will take charge of the United team for Tuesday's Champions League match away to Villarreal.
It was ultimately a disappointing derby with City's superiority in all areas confirmed abundantly while Ole Gunnar Solskjaer's showed once again they simply lack enough quality to mount even a token title bid.
The 36-year-old Ronaldo has now scored five goals in four games for United in this Champions League campaign and has nine in 11 outings for the club this season.
United came into the contest on the back of a 5-0 hammering by their bitter rivals Liverpool but goals from Ronaldo, Edinson Cavani and Marcus Rashford sealed a comfortable win.
Liverpool romped to a stunning 5-0 victory at Old Trafford that leaves Ole Gunnar Solskjaer's future in serious doubt.
After a poor first-half display, with Atalanta taking a 2-0 lead in at the break, Ole Gunnar Solskjaer's team turned the game -- and possibly their season -- around with a thunderous performance.
The result also ended United's record 29-game unbeaten away run in the league as they collapsed in a frantic five-minute spell in the closing stages.
Everton had a goal ruled out for offside by VAR after Townsend cancelled out Martial's opener.
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