Bale signed one of the most lucrative deals in football history on Monday to keep him at the European champions until 2021.
The Welshman earned his bumper new wage as he struck twice in seven first-half minutes before Alvaro Morata added a third 14 minutes from time.
“An easy game doesn’t exist and you saw that the first half was difficult for us,” said Madrid manager Zinedine Zidane.
“Once you score a goal it is a different game and we controlled the match much more in the second-half.”
Bale also scored Real’s fastest ever Champions League goal in midweek against Legia Warsaw, but his fine strike in the Polish capital was forgotten as Real blew a two-goal lead to draw 3-3.
There was to no such comeback from La Liga newcomers Leganes on their first ever league visit to the Santiago Bernabeu after Bale had awoken Madrid from their slumber in the first 30 minutes.
Bale outsprinted the Leganes defence as he latched onto Isco’s through ball and rounded Jon Ander Serantes before slotting home from a narrow angle.
His second was a scrappier finish as Bale pounced on a loose ball inside the area from Toni Kroos’ free-kick to bundle home from close range.
Serantes prevented Bale completing his hat-trick twice within a minute at the start of the second period.
Bale’s decision to go alone with his first effort infuriated the off-colour Cristiano Ronaldo, who was waiting for the ball to be squared.
Spanish media reports on Sunday suggest Ronaldo is on the verge of sealing his own mega contract extension at Madrid.
However, he endured another frustrating afternoon as he was kept scoreless for the fifth time in eight La Liga appearances this season.
The Portuguese has now gone five games without scoring at the Bernabeu – his longest drought since joining Real seven years ago. He was also booked early on for taking his protests towards referee Miguel Mateu Lahoz too far.
“There is nothing wrong, he is not worried,” said Zidane about Ronaldo’s slump in form.
“He looks good, he is motivated, he likes to scores goals and even more so in this stadium, but he is calm.”
Real received a huge boost ahead of a hugely demanding two-month stretch after the upcoming international break as Luka Modric made his return as a second-half substitute after a month out due to knee surgery.
Morata rounded off a comfortable afternoon for Zidane’s men when he collected Kroos’s pass before firing home at the near post.
And Madrid even kept their first clean sheet in 11 matches as victory stretches their unbeaten run in all competitions to 28 games.
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