Short answer, no.

Many of the Central Asian Khans did drink. In fact, before Babur moved to India and was a young prince dreaming of conquering Samarkand, he would go to visit his distant relatives in Herat who would host what he described as marvelous parties of drinking, poetry, dance, and music. Although he was enamored by the lifestyle of Herat, he saw himself as a military man and instead opted to leave his cousins in search for a Kingdom of his own.

It was part of Safavid court life to drink at dinner and to hold aristocratic parties.

Mughal Empire

Babur drank wine and enjoyed it immensely. He made a vow before the Battle of Panipat to give up on drinking alcohol in order to rally his troops as warriors of God as a means to raise morale before they battled the Lodi Dynasty. Babur would regret this vow later on and would write in his autobiography Baburnama about how much he misses the sweet taste of wine. He would later break his vow and would begin drinking again.

Akbar did drink wine but gave up on drinking and eating meat after coming under the influence of Hindu holy men.

Jahangir was a heavy drinker. He drank a lot, he drank with his wife Nur Jahan, and he writes about drinking and taking opium in his autobiography Tuzk-e-Jahangiri (Jahangirnama). He wrote poems about drinking and considered himself to be a wine connoisseur.

Ottoman Empire

The earlier emperors such as Osman, Murad, and Bayezid were fairly devout and its not known if they drank alcohol (as far as I know and I'm open to correction). However, there were several Sultans that did drink nominally. Its suspected that Selim II was drunk when he slipped and cracked his head in a bath that lead to his death. Take it with a grain of salt since no one really seemed to have liked him and many a myth were formed around him, but he did enjoy drinking.

Sources:

Baburnama

Tuzk-e-Jahangiri

The Mughal Throne by Abraham Eraly

Osman's Dream: A History of the Ottoman Empire, 1300-1923 by Caroline Finkel

At least when it comes to Turkey (I can't speak to Central Asia), the answer is a big fat nope. Turks still drink alcohol, in fact, I was just watching a show on the Foods Channel (or whatever it's called these days) about alcohol in Turkey, and a couple of specific drinks. IIRC, the host pretty much asked "If you're Muslims, why are you drinking this stuff?" And everybody pretty much said, "Because it's traditional".

I wish I remembered what the drink was called... I'll see if I can find it on the Wiki.

EDIT: This may have been it (it sounds right). EDITII: It might have been rakı too... Regardless, the point is is that Turks definitely didn't stop drinking.