The NET Bible makes an interesting point at footnote 54:
Most read your sons as a reference to Jewish exorcists (cf. “your followers,” L&N 9.4; for various views see D. L. Bock, Luke [BECNT], 2:1077-78), but more likely this is a reference to the disciples of Jesus themselves, who are also Jewish and have been healing as well (R. J. Shirock, “Whose Exorcists are they? The Referents of οἱ υἱοὶ ὑμῶν at Matthew 12:27/Luke 11:19,” JSNT 46 : 41-51). If this is a reference to the disciples, then Jesus’ point is that it is not only him, but those associated with him whose power the hearers must assess. The following reference to judging also favors this reading.
(L&N refers to Louw, Johannes P., and Eugene Nida, eds. Greek-English Lexicon of the New Testament Based on Semantic Domains. New York, NY: United Bible Societies, 1988. JSNT refers to Journal for the Study of the New Testament)
In short, if Shirock (in JSNT) is correct about Jesus' your sons referring to his disciples and not the Pharisees' disciples (see Matthew 12:27 for the identification of Jesus' accusers as Pharisees), then Jesus' argument--to me--makes more sense. Jesus' disciples, after all, were chronologically the Pharisees' disciples (i.e., "sons") before they were Jesus' disciples (i.e., "sons"). That would not mean that Jesus' disciples were also Pharisees; it would mean that his disciples (and Jesus, for that matter!) were brought up in Judaism and under the umbrella of Jewish rabbis, teachers, and leaders within the Jewish faith, regardless of their "party" affiliation.
Jesus' reasoning in both Matthew and Luke can be summarized as follows:
If I in fact am casting out demons by the power of their prince, Beelzebub [viz., Satan], that must mean Satan is working against himself, and that does not make sense.
Moreover, if I have given your sons who were brought up under your tutelage but are now my sons the power to cast out demons--and they have done so successfully, then by whose power do they do so: mine or Satan's?
If the Pharisees were to have answered Jesus' question at this point, they would have been caught on the horns of a dilemma! If they said, "Our disciples (i.e., our sons) also cast out demons by the power of Beelzebub," then they (the Pharisees) would be guilty of bad teaching and are just as much to blame as Jesus!
If, however, the Pharisees' former disciples (i.e., their sons) who are now under the authority of Jesus and like Jesus are casting out demons by the finger of God (Luke 11:20), then they will wind up judging the Pharisees.
Moreover, if that be the case, then the kingdom of God which Jesus preaches has not only arrived on the scene but has also surpassed the kingdom with which the Pharisees are aligned. In other words, Jesus turned the Pharisees' argument back onto them, with the implication that they were aligned with the wrong kingdom!
Jesus was therefore accusing the Pharisees of being aligned with Satan's purposes, just as Pharaoh's magicians were in Moses' day, since by their admission only the finger of the one true God could perform the miracle of the gnats (Exodus 8:19).