Real have underperformed on the domestic stage, trailing Barcelona by 15 points and knocked out of the King's Cup by Leganes, and many predicted an early elimination from Europe when they drew the runaway Ligue 1 leaders in the last 16.
Not for the first time, however, the 12-times European champions came into their own on the biggest stage of the continent.
The double defending champions held a 3-1 lead from the first leg and, aided by the absence of Neymar, nullified the French side in Tuesday's second leg amid the billowing smoke of flares at an intimidating Parc des Princes to win 2-1.
"I have been here for three years and we always play better in the Champions League. Madrid were just to born to win," said Kovacic, who is targeting a third consecutive winners medal in as many campaigns with the side.
"It's always great when you win a game as important as this. PSG played well, it was difficult for us but we managed the tie very well. They had more possession and played tough, they ran a lot, but we are also very tough to beat."
Real talisman Cristiano Ronaldo is the all-time leading scorer in the Champions League with 117 goals and in opening the scoring with a header, he matched Ruud van Nistelrooy's record of netting in eight consecutive games in the competition.
Edinson Cavani's rebounded effort gave 10-man PSG faint hope of a belated comeback but Casemiro's deflected goal at the other end, coming after Marco Verratti was sent off for the hosts, secured Real's passage into the last eight.
"We knew this was the perfect stage for us to vindicate ourselves in our favourite competition," added Real captain Ramos, who has hoisted the famous trophy three times.
"We dominated the game by enjoying our football again, we've recovered that state of mind which has helped us achieve so much and I hope this continues because we still haven't won anything."
(Reporting by Richard Martin; Editing by John O'Brien)
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