A brace from Nikica Jelavic and goals from Tom Huddlestone and Jake Livermore condemned the Bluebirds to their sixth home loss in the Premier League this season.
With just one win in 11 league games, Solskjaer admits that his side were below-par whilst reserving praise for the Tigers.
"We lost 4-0 and no matter how many chances you create and shots you have, you end up with nothing. It's an embarrassing defeat," he said.
"I'm not going to make any excuses. We just weren't good enough.
"We created quite a few decent chances, but that's football. It's all about scoring goals, taking your chances and defending well - and that's what they did. It was a really efficient display from their side and they defended really well."
The Norwegian claims that he has not underestimated the Premier League and has vowed to continue in his mission to save Cardiff from relegation, insisting that five wins should keep them in the league.
"I've not seen (the team) this low. It's a monumental task and it's the toughest challenge I've had in my career. It's not been a long career but it's going to be a challenge," he said.
"This result makes it harder but up until now I've not been surprised by the level in the Premier League. I still think we played some very good football at times and not got the results we should have had, but it counts for nothing.
"We have to win five games. That makes 37 points and that's what we need. It's at home you're going to get most of your points," he added.
"I'm not going to lay down. I've never quit in my life. I've been disappointed many times in my career, as a player and as a coach and as a manager, I'm not going to give up."