At least in India, War elephants were used well into the Late Medieval Age.

There are multiple reasons for the persistence of the War Elephant in India. For one, this is the native range of the Asian elephant, and the forests dotting the subcontinent were once teeming with elephants. War Elephants were necessarily captured from the wild, and not born into captivity. You do not want a domesticated elephant as a War Elephant. The ancient texts describe that War Elephants should be in musth, when bull elephants are in "heat", and are highly aggressive and barely controllable. You need access to a large pool of eligible bulls to be able to field hundreds of War Elephants.

Another factor in the persistence of elephants in combat were the relative shortage of good war horses in India. Early Indians were quite familiar with the Steppe cavalry techniques, but these horses couldn't be successfully bred in India. The horses were imported from Afghanistan, which restricted the sizes of Indian cavalries. Indian combat horses like the Marwari and the Kathiawari horses were developed only in the 12th century by crossing with Arabian horses.

As the Indian horses were developed, the size of the cavalry increased, decreasing the dependence on the War Elephant. Still, elephants were a formidable part of any medieval Indian military. Tamerlane's invasion in the late 14th century required special preparations to deal with the elephants of the Delhi Sultan. These included dropping caltropes into the path of advancing elephants, and setting fire to charging pack animals on the battlefield, spooking the elephants.

The nail in the coffin of the War Elephant was the arrival of cannons shortly after.