United had struggled to create clear chances against Patrick Vieira’s well-drilled side but the oft-criticised Fred beat Vincente Guaita with a wonderful strike from the edge of the box after being set up by substitute Mason Greenwood.
Palace should have gone ahead minutes earlier but Jordan Ayew missed from point-blank range at the back post after James Tomkins had headed down a corner.
United moved up to sixth place in the table on 24 points, three behind fourth-placed West Ham United. Palace slipped to 12th on 16 points.
German Rangnick was leading United for the first time after taking over from caretaker manager Michael Carrick on Friday and made a low-key entrance to his new home ground, with no announcement of his presence.
Former RB Leipzig coach Rangnick has promised to turn United into “pressing monsters” in the manner of Juergen Klopp’s Liverpool but says that approach will take time to instil in the players.
There were clear signs, though, of that shift in emphasis in the opening 25 minutes as United’s forward players closed down the Palace defenders and limited their ability to play out from the back.
Rangnick opted for a formation with Marcus Rashford and Cristiano Ronaldo in attack, with Jadon Sancho and Bruno Fernandes operating behind them.
Portuguese midfielder Fernandes had a chance to open the scoring after an excellent header back into his path from compatriot Ronaldo but his mis-hit shot was saved by Guaita.
Diogo Dalot, again preferred to Aaron Wan-Bissaka at right back, worked an opening just before the break, but fired his left-foot shot high and wide.
United struggled to build on the promising start, however, losing their way after the break with sloppy passing and a drop off in their movement and pressure.
But Rangnick will be pleased his team limited Palace’s effectiveness on the break with dangerous winger Wilfried Zaha making little impact.
United left back Alex Telles clipped the bar with a dipping free kick from a tight angle before Ayew wasted his golden chance to beat David De Gea.
Then came Fred’s moment of inspiration and the defensive midfielder, whose limitations have drawn plenty of criticism during United’s poor start to the season, was mobbed by his teammates in celebration.
“Fred has these moments, one shot per year. Everyone is happy for him because he’s such a nice guy. He works a lot and people don’t talk about him. He deserves the praise,” said Fernandes.
Vieira felt his team had deserved more.
“When we had that chance with Jordan, we didn’t take it. Against the big teams you get punished. In these games you have to score those,” he said.
“With their new manager we didn’t know how they would play. But we prepared with our shape. When you look at their chances, I think a draw would have been a fair result.”