Having established himself as Genk's first-choice goalkeeper when only a teenager, the Blues paid a reported £7.9million to sign the highly-rated Belgian in 2011.
Courtois has yet to feature for Chelsea three years on having been farmed out on loan to Atletico, where he has established himself as one of the best young goalkeepers in the world.
Chelsea found that out first-hand on Wednesday as the 21-year-old helped knock his parent club out of the Champions League at the semi-final juncture, with Jose Mourinho highlighting his save to deny a second-half John Terry header as the key moment given Diego Costa netted moments later.
Opposite number Schwarzer went and congratulated Courtois after Atleti's 3-1 win and the 41-year-old believes Chelsea have an exceptional player on their books.
"He is very special and it is no surprise that Chelsea signed him a couple of years ago," said the Australian, who made his first-team bow two years before Courtois was born.
"There has been a lot spoken about him and he has been outstanding
He was outstanding (against us) and he has a huge, huge career ahead of him.
"If we'd been a bit more lucky, and him a bit unlucky, we could have possibly got back into the game and created a few more nervy moments at the end for them
It wasn't the case.
"He did very well and it was no surprise that Chelsea signed him and want to keep him at the club
I wouldn't be surprised if they give him a new deal.
"He's had a fantastic season, as has Atletico
They're top of La Liga and vying to win the championship and now the Champions League final too."
Schwarzer became the oldest player to ever start a Champions League knockout tie against Atleti, playing after an injury to Petr Cech.
The 31-year-old Cech has been undisputed number one at Stamford Bridge for the past decade, but Courtois' fine displays is making that position look weak for the first time.
"It's a very tough decision and I'm glad I'm not the one that has to make it," Schwarzer said with a smile.
"I don't have a clue what they will decide it on
They are both world-class keepers.
"I wouldn't even say they are at different ends of their careers because Pete still has a long way to go in his
He can improve at 31."
Cech is likely to miss the rest of the campaign with a dislocated shoulder sustained in the first leg, meaning Schwarzer will again be between the sticks when Chelsea turn to domestic action this weekend.
Mourinho has given his side Friday off ahead of the penultimate game of the season against Norwich on Sunday - a match they approach two points off the top after their 2-0 win at leaders Liverpool last weekend.
"We need to put the others under pressure," Schwarzer said.
"We put ourselves back in the race with the win at Liverpool and we have to make sure we maintain it
We could still have something to play for in the last game at Cardiff.
"There is still a chance so we can't be sorry for ourselves.
"We have to pick ourselves up from this disappointment and realise we must keep banging on the door with the game against Norwich on Sunday.
"We must make sure we pick up three points so we can capitalise on any slip-ups."
Source : PA
Source: PA