Minamino came off the bench to score in a 3-0 win for Jurgen Klopp's side.
Rangnick steps in to replace Solskjaer, who was removed after United's 4-1 defeat by Watford this month.
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.
Chelsea manager Thomas Tuchel had called on his attacking players to step up after injuries to striker Romelu Lukaku and Timo Werner.
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.
Kevin De Bruyne scored in the 81st minute to make it 2-2 and give City a point at Anfield.
The 41-year-old Spaniard became the first Premier League manager to lose his job this season.
Timo Werner, Trevor Chalobah and Ben Chilwell scored for the Blues.
The 21-year-old England international has started only two of Manchester United's five Premier League games since his move from Borussia Dortmund.
Sadio Mane, Mo Salah and Naby Keita scored for the Reds in the Premier League match at Anfield.