Second-half goals by substitute Romelu Lukaku, with his first touch, and Marcus Rashford accounted for the Magpies and ensured Solskjaer follows Sir Matt Busby in winning each of his first four matches at the helm.
A clean sheet will also delight the Norwegian as Rafa Benitez's side pushed hard but were unable to find a way past a determined Reds rearguard. With Chelsea held at home to Southampton, the battle to get a top-four slot seems very much alive.
United made three changes to the team that demolished Bournemouth 4-1 last time out, with skipper Antonio Valencia, Phil Jones and Juan Mata replacing Ashley Young, the suspended Eric Bailly and Jesse Lingard.