Solskjaer became United's interim manager in December, replacing Jose Mourinho after a 3-1 defeat to Liverpool.
The Norwegian swiftly instilled a more attack-minded approach and United have remained unbeaten in the Premier League since he took over.
Sunday's goalless draw at home to Liverpool is only the second time they have dropped points in the league under Solskjaer – the other being against Burnley at Old Trafford, when United came from two goals down to rescue a point in a 2-2 draw.
The draw against Liverpool will certainly go down as a good result for United given the context, as the hosts lost Nemanja Matic to injury in the build-up, before also seeing Ander Herrera, Juan Mata and Jesse Lingard all forced off in the first half.
Marcus Rashford also struggled for most of the match with a knock sustained early on.
Solskjaer overtakes Guus Hiddink, formerly of Chelsea, for the most points won by a manager in their first 10 Premier League games – the Dutchman having taken 25.
The fixture also saw David de Gea keep a clean sheet for the 100th time in the Premier League, making him the second goalkeeper to do so for United after Peter Schmeichel.
Untied are back in action on Wednesday as they take on Crystal Palace in a midweek Premier League tie at Selhurst Park.