The Germany international joined the European champions from Bayern Munich for a reported €25 million (£20m) transfer fee earlier this summer and started in midfielder alongside Luka Modric in his new club's league opener at the Santiago Bernabeu.
Madrid toiled for long stretches of the game but Kroos showed signs of developing a good understanding with his Croatian colleague and also ended with an assist to his name, having whipped over the corner from which Karim Benzema opened the scoring.
However, the 24-year-old was reluctant to accept any praise for an encouraging display on his debut in La Liga, revealing that his focus has already switched to their tricky trip to San Sebastian to take on Real Sociedad.
"I felt very comfortable on the pitch," Kroos told reporters. "This victory is important. The first match of the league is always tough and we managed to win.
"We've had a series of very intense matches. Now we have a week to prepare for the next one. The visit to Real Sociedad will be difficult but we will go all out for victory at the Anoeta."