The Saints goalkeeper earned his team a 0-0 draw against Norwich on Saturday by saving Grant Holt's spot kick in stoppage time and believes they will avoid the drop
Boruc denied Grant Holt with an excellent save to his right after Luke Shaw had been harshly adjudged to have fouled the striker in injury time.
And the Polish goalkeeper believes Mauricio Pochettino's team are playing well enough to retain Premier League status despite having won only one of their last eight matches.
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"We are quite confident. I think it's a matter of time and we are going to grab points.
"It's always important not to lose, especially today against Norwich, it's difficult. So we're pretty happy. We could have scored two or three goals, so we're hopeful the next game is going to be better.
"We're working hard and I think we've shown in the last couple of games that we can play good football. We can score goals as well."
The former Celtic player has established himself as the club's first-choice goalkeeper this season but would not be drawn on the possibility of extending his one-year contract.
"I'll leave it to my agent and the club," he added. "I'm just doing my best and I'm working hard, so we'll see what's going to happen.
Boruc's opposite number, Norwich goalkeeper Mark Bunn, expressed his sympathy for Holt after the striker came up short in his attempt to provide the winning goal for the second home game in a row.
"He was a bit down, but at the end of the day it's a fantastic save by their goalkeeper to keep them in the game," Bunn told reporters.
"A draw was probably a fair result, we weren't at our best. But we could have nicked it at the end and that would have sent us 10th in the league, so that's how tight it is."
Norwich are nine points clear of the bottom three but Bunn insisted that the Canaries were not taking their own survival for granted.
"We don't see ourselves as safe," he claimed. "I think we need a couple more wins and we'll be alright."