The Welsh side slipped deeper into relegation trouble as they suffered a heartbreaking 2-1 loss at Goodison Park after a last-gasp, mishit strike by Seamus Coleman.
Cardiff had looked good for a point after goalkeeper David Marshall produced several fine saves to repel the Merseysiders and Juan Cala cancelled out Gerard Deulofeu's deflected opener.
With just one win in their last six games, they face an increasingly tough fight to climb out of the relegation zone but Solskjaer retains strong belief.
The Norwegian said: "With this attitude we will get points
We will see how many we need, but I can't say enough how pleased I was with the performance.
"I thought we performed fantastically
Everton have been outstanding this season at home and we gave them a game.
"We get on the ball, we defend, put our bodies on the line and then they get a goal at the end.
"Of course we need points
We need points to get clear of the relegation zone.
"But it has turned
We have turned the corner in the last couple of weeks, the performances have been a lot better.
"We won last week and should have had points here as well
We're not bad.
"Every game is a huge game now but if we have this attitude to the end of the season, we will be fine
The attitude is different class at the moment."
Everton dominated the contest and could justifiably claim victory was deserved after Marshall denied Kevin Mirallas, Deulofeu, Romelu Lukaku twice and Ross Barkley with top-class stops.
It took a deflection off Steven Caulker after 59 minutes for Deulofeu to finally break through.
But Cardiff battled and got back into the contest nine minutes later when Cala bundled in a Peter Whittingham free-kick.
Defeat therefore came as a cruel blow for them.
The visitors also felt they should have had a penalty when Wilfried Zaha went down under a challenge from Sylvain Distin shortly before Coleman sliced a shot that looped into the top corner in injury time.
Marshall said: "It was an unbelievable finish
The boy mishit the shot.
"It was a sickener but we showed a lot of character and played well
It was a good, positive performance.
"We're disappointed but we will take a lot of confidence from it.
"We have a home game against Liverpool next week but our stadium is a hard place to come to.
"If we start well we know we can get a positive result
Then the games after that are the teams round about us, so we can take confidence."
For Everton, the result was a boost to their push for European qualification.
Defender John Stones said: "It was a big relief but I think we knew it was coming.
"We would have liked it to come sooner but Seamus got there in the end.
"It took a long time to get in the net but I think we deserved the three points.
"Cardiff had a really good go at it and put us under quite a lot of pressure but overall I think we were the better team and deserved that second goal."
Everton seem to be in a battle with Tottenham and Manchester United for fifth place but no-one at the club is ruling out the possibility of snatching fourth and a Champions League spot.
Stones said: "We take each game at a time but never say never
We have got great character in there and I am sure we can achieve something special this season."
Stones, 19, is enjoying a promising first full season with the Toffees and has played the last three games at centre-back in the absence of England's Phil Jagielka through injury.
Stones, signed from Barnsley in January last year, said: "I am loving every minute of it.
"The more game time and the more experience I get from the players, playing against Premier League opposition, it only benefits me.
"Jags has been injured but he is still putting his influence on me in training, giving me advice
So is Sylvain.
"It is all looking forward and I can't thank them enough."
Source : PA
Source: PA