United were criticised in some quarters for their set-up in the 0-0 draw, with former Liverpool defender and Sky Sports pundit Jamie Carragher questioning whether Solskjaer's side showed the mentality of a title contender.
Carragher argued that United failed to show "authority", despite having two late chances to take all three points through Bruno Fernandes and Paul Pogba.
Solskjaer has an opportunity to secure his first win over Liverpool as United manager at Old Trafford this weekend in the FA Cup fourth round but believes that his approach at Anfield last Sunday was justified.
"We attacked every time we got the ball, maybe too much," he said. "We kept giving it away, that was the challenge. We went forward too quickly.
"The thought that we weren't attacking, for me I don't understand that one. We kept giving it away. It's the quality of what we did, we need to do better.
"We tried to press them high. It's not easy, they played well, they played their way out of our press. We didn't go out there to drop deep and try to soak up the pressure.
"They had five midfielders out there to play out plus a 'keeper who is very capable. When you look back at it, I think it was a good game by the two teams.
"We played against a top Liverpool team. Yeah, Van Dijk is not there but everyone's got an injury or two here or there. I was very pleased with the performance when I looked at it after."
Since his appointment, Solskjaer has looked to play a style that gets the best out of forwards like Marcus Rashford and Anthony Martial.
Rashford was caught offside five times at Anfield, the most of any Premier League player in a single game this season and the most for a United player since Robin van Persie against Tottenham in 2013.
United's quick transitions have led them to the top of the table but Solskjaer believes that more fine-tuning is needed in order to get the best out of his attack.
"You can do loads of work and we do loads of work but it's about taking a moment in the game and I feel we're getting there," he said.
"It's always been in the DNA of United that we do attack quickly. We do go fast. We have quick and fast players who can get up the pitch quickly.
"When you get up the pitch, the decision might be to slow it down and play in their half, not just going for the killer pass. They're the little things we're trying to fine-tune, of course."