The Red Devils had to defend deep and resolutely at Anfield, but did create opportunities on the counter attack of their own.
While a point keeps them top of the Premier League at the halfway mark, Solskjaer knows his team can and must do better to sustain a title challenge.
Asked about areas for improvement, he told the club website: "We can be more composed, we can have the ball more, we had to defend really often and I thought we gave the ball away too often.
"Maybe some moments we missed, when Marcus [Rashford] makes his runs in behind – maybe sometimes we don't pass it every time he runs and sometimes we missed it when he should have had it.
"So a couple of things that we need to improve on."