France midfielder Pogba was recalled to the starting line-up after missing the defeat at Manchester City two weeks ago with a muscle problem.
But Pogba lasted just 67 minutes before he was substituted in what was a flat United display, with the team jeered by frustrated fans at the final whistle.
Jose Mourinho's side remain well adrift of the Premier League's top four as they enter arguably the most intense period of the season, with seven more top-flight fixtures to come before the end of December.
Mourinho had suggested before the match that United were capable of bridging the gap between themselves and the Champions League qualification places.
However, they laboured, with Romelu Lukaku failing to score for the 11th club game in a row, despite having a goal disallowed for offside 12 minutes into the second half, when he touched in after goalkeeper Wayne Hennessey had pushed out Ashley Young's shot.
The Belgium striker then saw a well-directed header pushed aside by Hennessey in the final 10 minutes as United chased a winner.
Palace, who went into the game without a victory in seven league matches, had already seen a goal disallowed for offside by then, with Cheikhou Kouyate flagged narrowly offside as he touched in Luka Milivojevic's free-kick with his shoulder shortly before half-time.
Milivojevic was involved in the game's other major talking point, as the victim of a reckless challenge in the final 25 minutes by substitute Marouane Fellaini, who escaped without even a booking.
Pogba had a disappointing afternoon as Mourinho's attacking selection failed to bring reward, although he was not alone in that.
Despite a starting line-up that included Juan Mata, Jesse Lingard and Anthony Martial, United lacked the speed of thought to break down their opponents.
Indeed, Palace would have won a league match at Old Trafford for the first time since 1989 had Andros Townsend not steered wide after a bludgeoning run at the United defence with little more than 15 minutes to go.
Roy Hodgson's side had seen two good chances to score first come and go in the opening 25 minutes.
Wilfried Zaha, the former United forward, was involved in both opportunities, drilling the first wide from 20 yards after Lingard's sloppy pass out of defence had allowed Max Meyer to play the ball through.
The second chance was by far the better of the two, though, with Meyer running at the United defence and playing a diagonal pass to the right for Zaha, who then fed the ball left for the onrushing Patrick van Aanholt; the full-back stubbed his shot wide, the frustration clear on his face.
Lingard had United's first chance, with a shot from Lukaku's knockdown that was blocked by Aaron Wan-Bissaka, who then made a timely intervention near his own goal to prevent Martial from getting Nemanja Matic's lofted pass under control.
He was involved again as Hennessey saved Palace twice in quick succession on the half-hour mark, first pushing away Lingard's sliding shot from Young's low cross, then holding Martial's tame header after the midfielder had hooked the loose ball back into the middle.
In between, Chris Smalling had United's clearest first-half chance, heading wide at the far post from Mata's floated free-kick.
Yet Palace continued to threaten after Lukaku's disallowed goal, with David de Gea making a double save, first diving full length to his right to push aside Kouyate's shot from the edge of the area, then blocking Van Aanholt's hesitant follow-up.
The introduction of Marcus Rashford and Sanchez did increase United's threat, but not enough ultimately to find a goal.