The elephant, christened Bangabahadur (Hero of Bengal), died on Tuesday at Sharishabarhi Upazila's Koyra village in Jamalpur, media reports said.
The Sharishabarhi Upazila Chief Executive formed a three-strong committee to conduct the elephant's autopsy, bdnews24 reported.
A veterinary surgeon told reporters, "Bangabahadur died of heart failure triggered by insufferable heat."
The bull elephant, weighing some five tonnes, was laid to rest at Koyra after the autopsy.
The elephant was washed across the border from Assam in the strong currents of the Brahmaputra and had entered Bangladesh through Kurhigram border on June 26.
It travelled a few hundred kilometres into Bangladesh to Sirajganj via Gaibandha and Jamalpur and then travelled back to Jamalpur.
On August 11, authorities managed to rescue it after tranquilising it, and tied it with ropes and shackles to stop it from running away again.
However, the elephant broke away from the shackles and strayed to a muddy farmland where it stayed till Tuesday. All attempts to move it from there went in vain, and it died.
Locals allege the elephant died due to negligence of the forest department officials.
Tapan Kumar Dey, a former Forest Conservator who led the rescue attempts, said, "We tried our best to rescue Bangabahadur and keep the elephant alive. Its death has pained us."