Answer

In the Name of Allah, Most Gracious, Most Merciful. 

All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.

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In this fatwa:

There is no harm in reading from a copy of the Quran during the late-night [tahajjud] prayer.

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Answering your question, the late Saudi scholar, Sheikh Ibn Baz (may Allah have mercy on him) stated: 

There is no harm in reading from a copy of the Quran during tahajjud prayer. Dhakwan, the freed slave of Aishah, used to lead them in prayer during Ramadan reading from a copy of the Quran.

Al-Bukhari recorded that in his Sahih with a definitively-stated discontinuous chain. This is the opinion of a large number of scholars.

Those who don’t allow it have no evidence to support them.

Is Purification a Condition for Touching and Reading the Glorious Quran?

Not everyone has the Quran memorized and, therefore, there is a need to read it from a copy of the Quran in the prayer and otherwise.

This is especially true for the late night prayers and during the night prayers of Ramadan for those who have not memorized the Quran in their hearts.

Almighty Allah knows best.

Source: Islamic Fatawa Regarding Women, Compiled by Muhammad Ibn Abdul-Aziz Al-Musnad and translated by Jamal Al-Din Zarabozo, Darussalam, 1996.