The Mongols invaded India for the first time during the reign of Iltutmish. Iltutmish was the third ruler of the Delhi Sultanate and his reign lasted from 1211 to 1236. The Mongols, led by the famous Mongol warrior Genghis Khan, began their military campaigns in the early 13th century. They rapidly expanded their empire across Asia and had already conquered Central Asia before turning their attention towards India.

During Iltutmish`s reign, the Mongols launched multiple invasions into India. They primarily targeted the northwestern region and sacked cities like Multan and Lahore. However, Iltutmish successfully defended his capital, Delhi, and repelled the Mongol forces. This marked the first encounter between the Mongols and the Delhi Sultanate.

It is important to note that the other options provided in the question are not accurate. Allauddin Khilji, Balban, and Firoz Shah Tughlak were later rulers of the Delhi Sultanate and did not face the Mongol invasions for the first time. Therefore, the correct answer is option 4, Iltutmish.