For Naruto to be able to use Kurama's chakra he had to go through the whole chakra tug of war thing with the help of Killer Bee and Gyuki, but after Minato gets to the war he automatically is able to connect with Yang Kurama and transform. When did he learn to control Kurama's chakra?
It seems to be Kurama's Yin half within Minato noted that Naruto's actions and speech about his father was touching enough to move this half of it. This allowed the fourth to use Kurama's chakra. There was no need of tug war since Kurama willingly cooperated. It was not planing to take over anybody body or break free.
Re quote from the link above from the first paragraph of Birth of the Ten-Tails' Jinchūriki:
Meanwhile, as their battle strategy unfolded, Kurama's Yin half within Minato noted that Naruto's actions and speech about his father was touching enough to move this half of it. Yin-Kurama also told Minato to disregard Obito's words as he was not responsible for any of those tragedies and that instead of leaving his son to do it, he should also act. Later, when Naruto and Minato bumped fists, Yin and Yang-Kurama came in contact with each other, with Yang-Kurama greeting its other half casually and asking it to share its chakra, which led Yin-Kurama to note that asking itself for chakra was an odd situation. Nonetheless, when the father-son duo prepared to deal with Obito, the two halves of Kurama also bumped fists as they prepared to initiate their strategy. As their chakra begin to meld, the two halves conversed about Minato and Naruto, leading Yin-Kurama to comment that Naruto must have had some upbringing to treat a tailed beast as a comrade.
Minato learned it when he entered in the midst of war to help Naruto. Of course, we all knew he's one of the most talented shinobi who ever lived. He used to seal the Nine-Tails into Naruto, which would allow small amounts of the Nine-Tails' chakra to leak out and naturally mix with Naruto's own.
Minato learned it when he was reincarnated. Unlike most Tailed Beasts, the Nine-Tails offers no resistance to cooperating with Minato, granting him immediate access to Nine-Tails Chakra Mode, which Naruto is impressed by. Minato's is slightly darker than Naruto's but is otherwise identical in appearance and abilities; he can use chakra arms, enter a Tailed Beast Mode, and create Tailed Beast Balls!
In my opinion minato learned to do it when his soul and the other half of kurama where sealed using minato's death seal . The third did mention something like both him and orochimaru fighting for eternity inside some realm when he used the death seal on orochimaru. Maybe minato learned to control that half of kurama during their "fight" in that realm.