The Welsh side were in front twice in the game at the Cardiff City Stadium, but couldn't get anything from the contest after a four-goal salvo from Brendan Rodgers's side in the second half.
Solskjaer praised his side's performance in the first half of the Premier League clash and claimed that his side's attentions have now switched to the key fixture against fellow strugglers West Bromwich Albion next Saturday.
"We will keep fighting," he told reporters. "We were fantastic in the first-half. When you play against an attacking side such as them you have to test their back-line, and I think that is what we did.
"We had spaces in a few places and got some fantastic goals. Liverpool didn't come up with much in the first-half, chances-wise.
"We faced the top scorers in Europe, with the teams around us looking like having more manageable games (on Saturday), and we are not worse off than we were at the start of the day. I think that is the biggest plus that has come out of it."
Cardiff sit three points from safety in 19th place in the Premier League table.