Roy Hodgson's side failed to score for the seventh consecutive league game as Jose Mourinho's side eased to victory at Old Trafford.
United went into the game as heavy favourites and they showed why with just three minutes on the clock as Marcus Rashford skipped away from Joel Ward down the left before pulling the ball back to Juan Mata on the edge of the box, with the Spaniard making no mistake, sweeping the ball past Hennessey and into the net.
Despite the early goal, it wasn’t as if United were creating a plethora of opportunities in the first 45, with large spells of the game passing without any real action of note.
Mamadou Sakho headed a free kick from the right over the bar but as the half wore on, Palace grew into the game somewhat and started to enjoy more possession.
A series of corners were headed away by Chris Smalling and Marouane Fellaini before Bakary Sako forced David De Gea into a save with a powerful effort from the corner of the box after being played in by Cabaye.
In the blink of an eye though, Palace were two down.
The visitors failed to close down Ashley Young on the left and he cut back onto his right foot and sent a curling cross towards the far post, where Fellaini was on hand to tap home from close range.
Jose Mourinho’s side threatened to extend their lead further before the break with Chris Smalling heading narrowly wide following a corner from the left, before Rashford was denied by the outstretched foot of Hennessey as he looked to find the net at the near post following Mata’s cross from the right.
United went into the break with a comfortable lead and just as in the first half, they were quick to find the net after the restart.
With just three minutes of the second period played, Fallaini once again found the net, this time glancing home Rashford’s free kick from the right.
Romelu Lukaku should have added another goal to his ever-increasing tally for the season soon after, but he was let down by a poor first touch when he looked to be in behind.
Palace deserved credit for the way they continued to battle despite the result being a foregone conclusion at this point as they continued their search for a first Premier League goal of the season.
Yohan Cabaye fired over from range before Sako lashed a half-chance over the bar from 12 yards.
that search was in vain though and United weren’t without chances to add to their lead as time ticked away.
Lukaku should have got himself on the scoresheet when he latched onto a loose ball after the referee played a good advantage following a foul on Lingard. The Belgian jinked past his marker and looked for the far corner but with Hennessey beaten, the ball flew just wide of the post.
Palace were indebted to Hennessey shortly afterwards as he produced a fine save to deny substitute Anthony Martial.
The Frenchman was put in down the left by a fine pass from Ander Herrera but the Welshman was off his line quickly to make the save.
There was nothing the Palace stopper could do with just four minutes remaining though as Lukaku finally got his goal.
Herrera and Martial combined once again down the left with the latter pulling the ball back for the Belgian to slot home from close range.
The win heaps the pressure on Manchester City, who must beat champions Chelsea in the evening's late kick-off if they are to go into the international break level with their rivals at the top of the Premier League.
Crystal Palace meanwhile, will simply be happy to not have to play next weekend.