Rangnick, who was announced as Ole Gunnar Solskjaer's replacement on Monday, was unable to lead from the dugout during Thursday's 3-2 victory over Arsenal, so Sunday marked his first official game in charge.
United's players showed they had taken their new manager's instructions on board as they pressed from the front and played at a noticeably quicker tempo.
The home side were by far the better team throughout as they closed down Palace at every opportunity and refused to allow their opponents any time on the ball.
Rangnick started with Cristiano Ronaldo in attack alongside Marcus Rashford with the two supported by Jadon Sancho and Bruno Fernandes, but United rarely threatened Palace goalkeeper Vicente Guaita with the final ball clearly lacking.
Palace had won on their two previous visits to Old Trafford, and as the game wore on and United's players were unable to press with the same intensity, causing some frustration among the home fans, there was a sense the visitors could make it a third consecutive success.
On 75 minutes, Ayew fired James Tomkins' knockdown across goal and wide with David de Gea beaten as Cheikhou Kouyate could not get on the end of it -- and they were made to pay for that miss as Fred opened the scoring just two minutes later.
The clean sheet was United's first at Old Trafford this season and it took them to three points behind fourth-place West Ham United, while Palace remain 13th.