Zinedine Zidane called for big year from Bale ahead of the match and the Wales worldwide responded by scoring Madrid's opening league goal of the season for a third consecutive campaign.
Deportivo continued to threaten, however, with Andone heading over the crossbar from close range in the 37th minute, while Bakkali was causing problems to the champions with his pace and trickery in a wide position.
Those squandered chances proved costly when Madrid moved ahead in the 20th minute.
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Kroos had the first half-chance of the second period after finding space in an advanced area, but the German's effort was deflected wide of the Deportivo post six minutes after the break.
But there was some frustration for Zidane, with Ramos, who had been fortunate earlier, dismissed in the second minute of added time after catching substitute Borja Valle with an arm in an aerial duel.
I don't know if Ramos' card is fair, I have to see it, but I'm not going to talk about that.
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Zidane made four changes from the side that completed a 5-1 aggregate hammering of Barcelona in the Spanish Super Cup on Wednesday.
Real Madrid ended the game as a contest just past the hour when Kroos found the back of the net with a powerful finish after Bale had delivered a low cross into the German worldwide.
- Keylor Navas has kept consecutive clean sheets for Real Madrid for the first time since November 2016 (all competitions). The winger scored one goal while setting up another, showing that he still has it in him to perform for Real.
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