Why is tobacco called Dohány in Hungarian and not Tütün?

When I visited Hungary in the summer, the language was totally alien to any other language I am remotely familiar with. Being from the Finno-Ugric group, it is remotely related to Finnish, and little else.

From the signs in the airplane, I learned that the word "dohány" means "smoke", i.e. tobacco. This was confirmed from shops, and even the name of a street in the Jewish section of the city.

The word is almost identical to the Arabic word Dukhan (دخان) which is classical Arabic for smoke. It is even the name of a chapter in the Quran (44), named in verse 10 of that chapter.

What is strange is that there is no way Arabic directly influenced Hungarian throughout history. The Arabs never invaded that part of Europe, nor there were significant trade links.

The use of tobacco did not spread in Europe until the 16th century, after it was brought from the New World, centuries after the Arab empires ceased to exist.

The only Oriental influence was via the Turks and the Ottoman Empire who ruled Hungary for close to two centuries. However, the word for tobacco in Turkish is Tütün and not Dokhan. The surname Tutunji توتنجي is common in the Turkish and Arab world today, meaning "[descendant of] tobacco merchant". Even Muhammad Ali of Egypt was known as Tutunji due to his profession prior to his military and political career.

So, how did tobacco ge the name dohány?