The home side had a couple of second-half goals ruled out by VAR, however, while both teams hit the post from free kicks in a lively second half.
Relive how the 90 minutes of action unfolded with Sports Mole's live text coverage below.
United beat Chelsea 4-0 in the reverse fixture on the opening weekend of the season and also picked up a 2-1 win here in the EFL Cup last 16 three months ago, but they are six points behind their opponents heading into this latest tussle.
STARTING XI: Caballero; James, Christensen, Rudiger, Azpilicueta; Kante, Jorginho, Kovacic; Pedro, Batshuayi, Willian
SUBS: Kepa, Tomori, Zouma, Alonso, Barkley, Mount, Giroud
STARTING XI: De Gea; Bailly, Maguire, Shaw; Wan Bissaka, Matic, Fred, Bruno, Williams; James, Martial
SUBS: Romero, Jones, Dalot, Mata, Pereira, Ighalo, Greenwood
New signing Ighalo is among United's subs having trained away from his new teammates for the vast majority of the winter break. Bailly comes in for a first appearance since April 2019 in place of the ill Victor Lindelof, while Nemanja Matic is given the nod in an injury-hit central midfield.
Wan-Bissaka sent in a good cross from the right and Martial got in front of Christensen to head in from the game's first on-target attempt.
Batshuayi missed Chelsea's best opportunity when side-footing wide from 12 yards.
CHELSEA SUBS: Kepa, Tomori, Zouma, Alonso, Barkley, Giroud
MANCHESTER UNITED SUBS: Romero, Jones, Dalot, Mata, Pereira, Ighalo, Greenwood
Giroud will be wondering exactly what he has to do to get some minutes on the back of his side's first-half display, while Barkley is another is currently out of the picture.
As for United, who were lucky not to have Maguire dismissed in the first half, they have Greenwood, Mata and Ighalo among their attacking options should Solskjaer wish to gamble.
The Blues had a couple of goals ruled out by VAR - one for Kurt Zouma and another for substitute Olivier Giroud - while both teams hit the post from free kicks.
This is a massive win for United on their return to Premier League action, a victory that takes them into seventh place and within three points of fourth-placed Chelsea.
The battle for Champions League football is well and truly alive!
If you would like to read our on-the-whistle match report, you can do so by clicking here. Goodbye for now.