From your question:
since [...], there isn't a readily apparent scriptural basis for it[.]
And Jesus said to them: Can the children of the bridegroom mourn, as long as the bridegroom is with them? But the days will come, when the bridegroom shall be taken away from them, and then they shall fast. (Matthew 9:15)
Jesus speaking of the Bridegroom, is referring to the wedding supper of the Lamb, the heavenly banquet, where at the time when he was speaking, he is not only physically present, but locally present, after his ascension, the Lord is not with us locally, we are not sharing a meal with him, but rather sharing him in a meal. (John 6 Bread of Life discourse)
Moses passed on that the ordinance of the Day of Atonement would be an on going feast. The Sacrifice of the Mass is a calibration of our Atoning sacrifice. In the similar manor we should fast.
And this shall be to you an everlasting ordinance. The seventh month, the tenth day of the month, you shall afflict your souls, and shall do no work, whether it be one of your own country, or a stranger that sojourneth among you. Upon this day shall be the expiation for you, and the cleansing from all your sins. You shall be cleansed before the Lord.
For it is a Sabbath of rest: and you shall afflict your souls by a perpetual religion. (Leviticus 16:29-31)
The Catechism of the Catholic Church has this to say
The interior penance of the Christian can be expressed in many and various ways. Scripture and the Fathers insist above all on three forms, fasting, prayer, and almsgiving, which express conversion in relation to oneself, to God, and to others. Alongside the radical purification brought about by Baptism or martyrdom they cite as means of obtaining forgiveness of sins: efforts at reconciliation with one’s neighbor, tears of repentance, concern for the salvation of one’s neighbor, the intercession of the saints, and the practice of charity “which covers a multitude of sins.” (CCC 1434)
Fasting, Prayer and Almsgiving are all part of the Catholic Mass.
Remember what it is..
St Augustine (d. 430) says: “It has pleased the Holy Spirit that, out of honor for this great sacrament, the Lord’s Body should enter the mouth of a Christian before other foods.”
It is fitting not to mix Christs flesh with a Bacon Egg and Cheese Biscuit.
When the Church prays, it prays as one. When the Church gives alms, it gives alms as one. When the Church fasts, it fasts as one. Although we sometimes pray separately, give alms separately and even fast for our own reasons, the Holy Spirit works through the one flesh union of its body. Our primary means and greatest form of Prayer and worship is the Mass.
The Mass is the Passover fulfilled. From ancient times it has been the Jewish Custom to celebrate a pre-Passover fast, The Fast of the First Born.
It is an ancient and widespread custom for the firstborn to fast on the day before Passover. This commemorates the miracle that spared the firstborn Jewish sons from the plague that struck down the firstborn sons of the Egyptians. Source (http://www.chabad.org/holidays/passover/pesach_cdo/aid/1678/jewish/The-Fast-of-the-First-Born.htm)
Is there scriptural support for fasting, yes, is there scripture support for the Eucharistic celebration and fasting, not specifically. The Traditions of the Church of God, which include the celebration of the Mass, are developed from Scripture the Law and the Prophets, ancient traditions and traditions passed on by Christ through his apostles and preserved and made clearer through guidance and direction the Holy Spirit.