The "finger of God" is mentioned in two passages in the Hebrew Bible. Once when the magicians of Pharaoh conceded defeat before Moses (Ex 8:19) and secondly when the two tablets were inscribed with the Ten Commandments (Ex 31:18).
In the context at hand here in Matt 12:27-28 and Luke 11:19-20, where we see Jesus casting out the demons, the confrontation hails back to the former episode in Exodus, when magicians in the court of Pharaoh were stymied by "the finger of God."
Thus, like Moses, Jesus had stymied the power of Satan, whom Jesus likens in this context to a strong man that is bound and plundered (Matt 12:29-30 and Luke 11:21-23). The strong man is stymied by "the finger of God." Please notice that Jesus makes the connection in the immediate context at hand.
According to Isaiah, the "Holy Spirit" of Yahweh was this power of God, which enabled Moses to lead the Israelites out of Egypt. The words are קָדְשֹׁו רוּחַ, which literally are translated as "his Holy Spirit" in Isaiah 63:10 and Isaiah 63:11. It was therefore "the Spirit of Yahweh" (Isaiah 63:14) who empowered Moses according to these three respective passages in Isaiah.
As we see in the Exodus account, the "mighty hand and outstretched arm" now come into focus in these passages from Isaiah. It was not the finger of God, but the "mighty hand and outstretched arm" of God that delivered them from Egypt. The progression of power starts with the finger, the hand, and then the arm. The visible theocratic Kingdom of God was subsequently established through the "mighty hand and outstretched arm" of God.
Jesus thus said to the Pharisees that "the Kingdom of God was nigh." He implied therefore to his listeners who were familiar with the Hebrew Bible that the "mighty hand and outstretched arm" of God was next, since the "finger of God" was now presently evident to them. In other words, when Jesus had mentioned that the Kingdom of God was at hand, he signaled the immanency of the return of the Kingdom of God to earth (just as the Exodus in Egypt resulted in the visible theocratic Kingdom at the giving of the Ten Commandments on Sinai). Please note that the Ten Commandments (Old Covenant) was given 50 days (Shavuot) after the Exodus from Egypt in the same way that the New Covenant was given 50 days (Pentecost) after the resurrection of Jesus the Nazarene.
Further, in Ezekiel 20:33-44, we see the predictive prophecy at the time that Ezekiel wrote, that the "mighty hand and outstretched arm" will again save the faithful remnant of Jews but through God's covenant, which was the New Covenant that was in view (cf. Ezek 20:37 with Ezek 11:19-20; Ezek 34:25; and Ezek 36:24-28). Therefore Jesus was heralding to his listeners the immanency of this New Covenant-based Kingdom.
The finger of God therefore prefigures "the mighty hand and outstretched arm" of God, who delivers his people through "his Holy Spirit" (קָדְשֹׁו רוּחַ). The Holy Spirit therefore is the power that binds Satan.
Jesus is the Moses who heralds the New Covenant-based Kingdom of God, which arrived at Pentecost by the Holy Spirit.