What is the biblical basis for classifying the Eucharist as an offering or sacrifice? It is commonly believed in Catholicism that the Eucharist/Lord's Supper/Communion is a "sacrifice" or "offering." 

For example, one website states,


  The Eucharist is a true sacrifice, not just a commemorative meal, as "Bible Christians" insist.


What is the biblical basis for this belief?

Note: 


"offering": προσφορά
"sacrifice": θυσία
Among other scriptures, the biblical basis for the Eucharist/Communion/Lord's Supper being an offering (προσφορά) or sacrifice (θυσία) is derived from 1 Cor. 10.
In 1 Cor. 10:18, the apostle Paul wrote,

Consider Israel according to the flesh. Are not those who eat the sacrifices partakers with the altar? 1
βλέπετε τὸν Ἰσραὴλ κατὰ σάρκα οὐχὶ οἱ ἐσθίοντες τὰς θυσίας κοινωνοὶ τοῦ θυσιαστηρίου εἰσίν

&quot;Israel according to the flesh&quot; refers to those Israelites physically descended from the patriarch Israel (Jacob) and his son Levi who participated in offering and eating the sacrifices at the altar in the Temple (cp. Deu. 18:1; Lev. 7:15-18).
The apostle Paul was concerned with the Corinthians partaking of heathen sacrifices offered to demons. By doing so, they would be partaking of the same sacrifices with demons.
In 1 Cor. 10:20, he wrote,

But [I say] that the things that the Gentiles sacrifice, they sacrifice to demons and not to God. But I do not desire for you to be partakers with the demons.
ἀλλ᾽ ὅτι ἃ θύει τά ἔθνη, δαιμονίοις θύει καὶ οὐ θεῷ οὐ θέλω δὲ ὑμᾶς κοινωνοὺς τῶν δαιμονίων γίνεσθαι

The apostle Paul describes how sacrificing to an entity causes one to be a partaker of that sacrifice with the entity. Ergo, because the heathens sacrifice to demons, they become partakers of that sacrifice with the demons. One of the fundamental beliefs concerning sacrifices was that the entity to whom the sacrifices were offered also participated in the sacrifice, this by means of the altar upon which the sacrifice was offered. The altar in Judaism, as well as heathen cultures who practiced sacrifices, represented the deity to whom the sacrifices were offered.
In Synonyms of the Old Testament: Their Bearing on Christian Faith and Practice, Ch. XVI, p. 310, Robert Baker Girdlestone wrote,

The altar, θυσιαστήριον, is mentioned in about twenty passages, in most of which the Jewish altar is referred to. In 1 Cor. 10.18, St. Paul reminds the Corinthians that in the case of Israel those who eat the sacrifices becoming in so doing partakers with the altar. By this he evidently means that while the altar (which represented God) had part of the victim, the sacrificer had another part; thus the sacrifice, being consumed partly by God (through means of the fire on the altar) and partly by man, forms a bond of union between the one and the other.

Finally, in 1 Cor. 10:21, the apostle Paul wrote,

You cannot drink the cup of the Lord and the cup of demons. You cannot partake of the table of the Lord and the table of demons.
οὐ δύνασθε ποτήριον κυρίου πίνειν καὶ ποτήριον δαιμονίων οὐ δύνασθε τραπέζης κυρίου μετέχειν καὶ τραπέζης δαιμονίων

The Greek word τράπεζα, translated as &quot;table,&quot; is a synonym for &quot;altar&quot; (θυσιαστήριον). For example, in Mal. 1:7, it is written,

You offer polluted bread 2 upon My altar. And you say, &quot;Wherein have we polluted You?&quot; When you say, &quot;The table of Yahveh is contemptible.&quot;
מַגִּישִׁים עַל מִזְבְּחִי לֶחֶם מְגֹאָל וַאֲמַרְתֶּם בַּמֶּה גֵאַלְנוּךָ בֶּאֱמָרְכֶם שֻׁלְחַן יהוה נִבְזֶה הוּא

In his commentary on Mal. 1:7, Franz Delitzsch wrote,

The table of Jehovah is the altar, upon which the sacrifices (i.e., the food of God) were laid.

Likewise, in his commentary on Mal. 1:7, Rabbi David Kimchi wrote,

&quot;The table of Yahveh&quot; - It is the altar, and so it said in Eze. [41:22] regarding the altar, &quot;This is the table that is before Yahveh.&quot;
שלחן יהוה - הוא המזבח וכן אמר ביחזקאל על המזבח: זה השלחן אשר לפני יהוה

Having established that &quot;table&quot; = &quot;altar,&quot; how does one partake of the table of demons (1 Cor. 10:21)? Evidently, it is by offering sacrifices to demons (1 Cor. 10:20). If one partakes of the table of demons by sacrificing to demons, how does one partake of the table of the Lord (i.e., table of Yahveh)? By analogy, one partakes of the table of the Lord by offering sacrifices to the Lord.
partake of the table of demons : offer sacrifices to demons ::
partake of the table of the Lord : offer sacrifices to the Lord
Of course, what are these sacrifices to the Lord? There can be no doubt that the apostle Paul is referring to &quot;the cup of blessing which we bless,&quot; that is, &quot;the communion of the blood of Christ,&quot; and &quot;the bread which we break,&quot; that is, &quot;the communion of the body of Christ&quot; which he mention only two verses earlier (1 Cor. 10:16). In other words, the cup and bread of the Eucharist are indeed sacrifices offered to God on His altar/table.

Footnotes
1 Henry Alford: &quot;in a strict and peculiar sense,—the altar having part of the animal, the partaker another part; and by the fact of the religious consecration of the offered part, this connexion becomes a religious connexion.&quot;
2 Franz Delitzsch: לֶחֶם, bread or food, does not refer to the shew-bread, for that was not offered upon the altar, but is the sacrificial flesh, which is called in Lev. 21:6, 21:8, 21:17, the food (לֶחֶם) of God