There is not a clear cause-effect relationship there, in this particular passage. Moreover, we learn that a strong faith that expels demons can be seen also in those people who are not pleasing Christ at all, that is to say, who are not practicing merciful deeds and do not even know God notwithstanding their miracle-working through invoking His name (Matthew 7:22).
Now, what is the presence of demon in us? It is presence of a fallen angel in us that uses our intellectual and bodily capacities for his pernicious agendum. But had this fallen angel not entered us, and had there been in us a presence of a sinful passion - without any fallen angel being present - that leads us to the same agendum, will any demon bother to enter in us? No, because we are already doing his agendum and he can take a gleeful rest and lay down on his demonic beach and drink his demonic pinna-coladas. Like in an old desert fathers' stories: two demons are walking in a desert, one experienced and another - novice; so, they see a drunken monk and the novice excitingly says to the experienced demon: "Let us seduce him and make a trick on him!" The experienced one retorts: "You, stupid, he is already ours, let us seduce the one who is sober, fasting and praying!"
So, not so much demons, but our sinful passions and inclinations are of the primary importance. Even if all demons are driven out from our galaxy, the humanity will not be any better or happier, for the sinful passions will remain all the same: they will have the same percentage and power on the earthly populations. Will not a plagiarist plagiarize without demons' seduction? Will not a naughty journalist write a rigged article for money without demons' seduction? Will not a womanizer lead a lewd life without demons forcing him to? Yes, yes and yes! So, it is important to fight passions and if you defeat passion, you have also defeated demon for demon grazes only on grass of our sinful passions and if we eradicate this grass through fasting and prayers, there will be nothing for demons to graze inside us.
Yes, one can expel demon and this act of expulsion will not make him, the expulsor himself any better person and closer to Christ, and contrary, one can fight one's sinful passions through prayers and fasting without working any miracle, and this will make him closer to Christ.
Eventually, of course, the one who is thus close to Christ by the reason of defeating one's sinful passions through prayers and fasting will be more entitled also to be granted by God the power to help others and also the power of expelling demons, if this last ability, which is a miracle-working ability will not make him turgid and arrogant. God usually wards even His chosen ones from miracle-working abilities in order to protect them from turgidness and arrogance. Because of this divine protection even Paul, the seer of the Third Heaven, was not granted expulsion of one naughty demon not only from others, but even from his own self, for so much God loved him and so much He wanted him to remain humble (2 Cor. 12:8-10), that He made even demon immune from Paul's prayers and let it remain in His "chosen vessel" (Acts 9:15).
Just imagine, how dangerously arrogant and self-aggrandized you yourself, who are now reading this, will turn, had you worked even one tiny miracle, if even Paul was not immune from this peril?!