Caulker's double helped Cardiff to their first win in five Premier League matches over their relegation rivals, a result that leaves them in the bottom three on goal difference alone.
The defender moved to Wales from Tottenham last July in order to play regular first-team football, and Solskjaer believes that decision could lead to the 22-year-old earning a place in England's FIFA World Cup squad.
"When he comes to Cardiff, he knows he's going to be fighting for survival and that's a big ask because he could have had an easy life being at Spurs probably," the Norwegian said.
"He took on a great test and he's maturing all the time and it's great to see.
"The lad is 22 years of age, he is a leader, he's very talented and he's had a testing season.
"He's also had to take on a lot of the leadership role because of Huds (Mark Hudson) being injured - especially on the pitch.
"He's first and foremost a defender, and Steven defended fantastically today. He'll chip in with goals, we are a threat at set-pieces and I think teams know that.
"I want him to focus on his performances here. If he plays like this, defensively, and he chips in with the odd goal, he's got to be in with a shout (of going to the World Cup)."
Solskjaer also called on everyone connected with Cardiff to pull together in their fight to avoid the drop, and insisted outspoken owner Vincent Tan only wants the best for the team.
"We're all here together," he added. "We've got to stick together to get the points we need and he's behind the team. He wants us to succeed.
"It's vital that the club and everyone attached to the club can focus on the matters on the pitch.
"The players want to focus on what they have to do every day in training.
"It's not about turning up every Saturday, they've got to do that every day in training.
"We've got to work hard, because it might be games you have to win by what you deserve."
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