Bruno Fernandes and Paul Pogba went closest for the visitors as Alisson Becker made two important saves to preserve a point for Liverpool, but Solskjaer was disappointed that the Red Devils could not make a title statement at Anfield.
"They are a good side and they have some injury problems but we didn't pounce on that," he said. "I felt we grew into the game and got stronger and stronger and were closer to winning.
"We were a bit disappointed in the performance, not just the result. We didn't do well enough to cause them problems in the first half but we defended well and they didn't create too many chances.
"We know we have players who can win you a game in the blink of a second and Bruno and Paul almost did.
"But I think everyone was a bit disappointed with the way we started the game but that is a good feeling to have that we were disappointed in the performance."
The draw leaves United three points ahead of third-placed Liverpool, though Leicester City’s win against Southampton on Saturday means Brendan Rodgers Foxes are just two points off top spot with a superior goal difference.
The result at Anfield was also a further boost to Manchester City, who can now go top provided they win both their games in hand.