United dominated possession and enjoyed more shots than Palace -- 11 in the first half alone -- but a goal proved elusive for much of the match.
However, Fred's strike from the edge of the box in the 77th minute settled the contest and moved United up to 6th in the Premier League table.
The goal came shortly after Palace had squandered its best chance of the match when Jordan Ayew couldn't turn a shot into the net following a corner.
United sacked manager Ole Gunnar Solskjaer two weeks ago, with Michael Carrick taking over as caretaker manager.
Rangnick, previously head of sports and development for Russian club Lokomotiv Moscow, was then named as the club's interim manager until the end of the season, after which he will stay at the club in a consultancy role for a further two years.
United is one point behind 5th-placed Tottenham Hotspur in the Premier League table. The north London side defeated Norwich City 3-0 on Sunday thanks to goals from Lucas Moura, Davinson Sanchez and Son Heung-min.
Rangnick said afterwards he was particularly pleased with United's first home clean sheet since April and singled out Ronaldo for his role in keeping Palace out.
