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In his 50th La Liga game as a Madrid player, goalkeeper Courtois made key first-half saves from Arthur and Lionel Messi which prevented the visitors from establishing a lead at Santiago Bernabeu.
That set the platform for second-half goals from Vinicius Junior and Mariano Diaz as Zinedine Zidane's men claimed a 2-0 home win on Sunday that put them top of La Liga.
Courtois felt Madrid's tactics made the difference and was glad to have stepped up when it mattered with important stops before the break.
"Against Barcelona you can play in two ways: get into the area and defend or press and risk as they play out," he said to the club's official website.
"We did the latter very well and winning a Clasico is always good.
"At 0-0 in the first half it was important not to concede any goals. When it was my turn I made good saves, like against Arthur and Messi.
"Then in the second half we were better and made the difference."
Now ahead of Barca by a solitary point in the domestic standings, he has urged his team-mates not to take their foot off the gas ahead of a crucial away fixture against Real Betis on March 8.
The Belgium international added: "We are ahead but we can't forget that there are 12 games left.
"Next we visit Betis and if we want to win LaLiga we have to win there too.
"After our last results, we have continued to work well. I think we played better in those games but we didn't win.
"Football is about small details and this time we got the win with the help of our fans. That is key and they have been incredible."
Five days after their trip to Benito Villamarin Stadium, Madrid are due to welcome Eibar to the Bernabeu, before preparations for the second leg of their round of 16 Champions League tie against City get underway.