Pardew is under pressure from the club's new owners after executive chairman Nicola Cortese recently claimed the play-offs are the priority despite starting the season with a 10-point deduction for going into administration.
Saints are currently 12 points shy of the top six and while Pardew has not given up on a return to Wembley at the end of the campaign, he at least has a trophy to show for his efforts so far.
"We have new owners who want success tomorrow which is fine but we know the football world doesn't work like that," Pardew said.
"But I got them a Wembley visit and we want to get to the play-offs and we think we've got a chance.
"I don't think it will buy me some time, I have to accept the pressure - he is being driven to get this team into the Premier League. His driving at me can sometimes, you could argue, put too much pressure on me. Sometimes it's just about right.
"But it was important we won because I can look him in the eye today and say, 'Look, we've done a lot of good things here'.
"We talked about two years in this division and I think I'll get this team out of it in that time.
"In three years I expect this team to be challenging to get into the Premier League."
Carlisle were in trouble from the moment Peter Murphy handled in the area and Rickie Lambert thumped home the penalty after just 15 minutes.
Adam Lallana nodded in the second before the break and Papa Waigo put the game beyond the Cumbrians early in the second half.
Southampton added a fourth through Michail Antonio before substitute Gary Madine gave the blue half of Wembley something to cheer.
"It's never nice to get to a cup final and lose but we contributed to our own downfall," said Carlisle boss Greg Abbott.
"We made individual errors and weren't good enough in each box. But Southampton came to win a cup and won it in style."