United, who only had one league win to their name in 2020 ahead of kickoff, scored once in either half in London, and the result has seen Ole Gunnar Solskjaer's side move into seventh in the table, just three points off fourth-placed Chelsea.
Chelsea, who were without Tammy Abraham and Callum Hudson-Odoi through injury, lined up with an unfamiliar front three of Willian, Pedro and Michy Batshuayi, while there was a spot in midfield for Mateo Kovacic as Mason Mount dropped out.
The Red Devils lined up in the the same 5-2-1-2 formation that had worked so well away to Manchester City in the second leg of their EFL Cup semi-final last month, but Bruno Fernandes and Daniel James were handed starts alongside Martial on this occasion rather than Mason Greenwood and Jesse Lingard.
Chelsea had a half-chance in the sixth minute when Reece James worked some space down the right before unleashing a strike, but the defender's effort was always moving wide of the post.
Willian then curled one over the United crossbar three minutes later as the home side continued to make ground down the United left.
United's first real sight of the Chelsea goal came from Fred, but the Brazilian's attempt, which came after smart combination work in the final third, was deflected behind for a corner.
Martial had a shot blocked in the 21st minute during a period where both sides were finding space between the lines, although neither goalkeeper was being worked.
Chelsea will feel that Maguire should have been sent off for a kick on Batshuayi shortly after, but neither the referee nor VAR felt that an offence had been committed.
Maguire then had to clear a dangerous Pedro cross behind for a corner, which almost led to a tap-in for Batshuayi, although Willian's cross towards the Belgian was just behind him as United escaped.
Chelsea had another huge chance in the 26th minute when Mount found Batshuayi free inside the United box, but the striker could only place his effort wide of the post.
Fernandes tried his luck for the visitors just past the half-hour mark after finding space in a dangerous position, although the Portugal international's effort was always missing the crossbar.
United had a big opportunity of their own in the 37th minute when Martial worked his way into the Chelsea box, but the Frenchman sent his effort wide of the far post.
Chelsea were continuing to make ground down the other end, but Bailly was impressing defensively as the Ivory Coast international bailed his side out of a number of dangerous situations.
United took the lead in the 45th minute, though, when a brilliant cross from Aaron Wan-Bissaka was headed into the bottom corner by Martial to spark jubilant scenes in the away stand.
Chelsea's best chance of a leveller fell to Batshuayi in the closing stages of the half, but the striker sent his effort wide of the far past as Solskjaer's side took a lead into the break.
There was a feeling among the home fans that Maguire should have seen red for his clash with Batshuayi, and Chelsea did start the second half on the front foot as they pushed forward in search of a leveller.
Mount forced De Gea into a smart save in the 54th minute, although the move was called back for an earlier handball, before Pedro had a shot deflected wide of the post moments later.
Chelsea thought that they were level from the resulting corner when Zouma found the bottom corner, but VAR found that an offence had taken place with Brandon Williams being pushed in the back by Cesar Azpilicueta before the French defender converted.
There was a real sense of injustice inside the stadium as the home supporters chanted against VAR, but United still had a one-goal lead heading into the final 30 minutes of action.
Fernandes came close to sending the visitors two goals ahead when he arrowed a free kick towards the bottom corner, but the post came to Chelsea's rescue.
The visitors did score a second in the 65th minute, though, when a brilliant corner from the excellent Fernandes was headed into the back of the net by Maguire.
Chelsea, who sent Giroud onto the field in the 68th minute, had a chance to answer back in the 73rd minute when Kovacic broke into the United box, but Bailly was on hand to make another fine challenge as the score stayed 2-0 to the visitors.
The Blues had the ball in the back of the net for a second time through Giroud in the 77th minute but once again VAR ruled it out for offside.
In the end, it just was not to be for the capital outfit, who hit the post late on through Mount before Solskjaer introduced Ighalo in place of Martial; it was almost the perfect debut for the striker when he was sent into a one-on-one with Caballero late on, but the experienced goalkeeper made a smart save.
Next up for United is a trip to Club Brugge in the Europa League on Thursday night, while Chelsea will welcome Tottenham Hotspur to Stamford Bridge in the Premier League on Saturday.
CHELSEA (4-3-3): Caballero; James, Rudiger, Christensen (Zouma 46'), Azpilicueta; Kante, Jorginho, Kovacic; Pedro, Batshuayi (Giroud 68'), Willian
MANCHESTER UNITED (5-2-1-2): De Gea; Wan-Bissaka, Bailly, Maguire, Shaw, Williams; Fred, Matic; Fernandes (Dalot 92'); James (Pereira 80'), Martial (Ighalo 90')