Defender Nacho added a second minutes later, poking the ball home after a well-worked corner.
But Granada did not fade away, and striker Luis Suarez pulled one back for the hosts after a period of sustained pressure.
Forward Vinicius Junior restored Madrid's two-goal cushion on 56 minutes, and Granada further struggled to close the gap after midfielder Monchu saw red in the closing stages before defender Ferland Mendy netted a fourth to cap his side's third straight league win.
"Really happy with how the game went," Asensio said in a postmatch news conference. "After an international break it's always difficult to get the dynamic back, but the team were serious, we played a great game and the most important thing is the three points."
Ancellotti told reporters after the match: "They [Luka Modric and Kroos] are still the best midfielders in the world for me. I have to say that the dynamic the three have surprises me -- they do things that I don't ask them to do but they do it naturally.
"It's a great advantage for this club, they have been the past and they are the present and future, in the short term. If we think about 10 years in the future we'll have to find something new, although we already have it because Camavinga and Valverde are ready and [Antonio] Blanco, too."