Stories that Cristiano Ronaldo was set to replace Harry Maguire as United captain for the rest of the season were dismissed as "nonsense" by Rangnick this week.
Maguire responded by opening the scoring on 34 minutes before Bruno Fernandes added a second on the stroke of halftime.
Lost leads have been a feature of Rangnick's interim spell in charge and United were punished for another slow start to a second half as Rodrigo's fortunate cross and substitute Raphinha brought Leeds level.
This time, United bounced back to find a winner when Fred smashed in 20 minutes from time before Anthony Elanga sealed the points late on.
"Today it was important not only that we won the game but the way we reacted. That was the best possible response the team could give," said Rangnick.
"A game like this one today you can only win as a team."
On their first league visit to Elland Road in front of a capacity crowd in 19 years, the visitors were met with a hostile atmosphere and a tricky pitch in incessant rain.
United managed to weather the first Leeds storm to take control of the first half.
The victory takes United to back into the top four position with 4 points between them and West Ham United.