How are people who were not approved by Jesus (Matthew 7:21-23) still able to expel demons?

tl;dr

It is possible that they are not casting out the demons but that the demons participate by leaving on their own accord, in order to provide credence to the preacher.

Exorcism exists also in other religions such as some forms of buddhism (they call it bgegs-bloz-chog or za-dre kha sgyur depending on the ritual) and among the nature religions of Africa. Sometimes the show to exercise the demon can be quite "extravagant". This is all for the audience because Satan's intention is to mislead, and confuse if necessary.

A longer explanation:

Demons are capable of other works as well that to us look miraculous, such as healing people.

Some of the miracles done by demons look like they are from God, such as in the case of the Simon mentioned in Acts 8:9-11. This passage says:

9 Now for some time a man named Simon had practiced sorcery in the city and amazed all the people of Samaria. He boasted that he was someone great, 10 and all the people, both high and low, gave him their attention and exclaimed, “This man is rightly called the Great Power of God.” 11 They followed him because he had amazed them for a long time with his sorcery.

To the people in the city of Samaria the miracles made by Simon looked like they were from God.

Expulsion of demons and other miracles were also accomplished through "magic" as in the case of people in the ancient city of Ephesus. Acts 19:18-19 tells us that the residents of Ephesus had a large amount of books of that nature:

18 Also many of those who were now believers came, confessing and divulging their practices. 19 And a number of those who had practiced magic arts brought their books together and burned them in the sight of all. And they counted the value of them and found it came to fifty thousand pieces of silver.

An old book titled "Notes, Explanatory and Practical, on the Acts of the Apostles" by A. Barnes (published 1863) said on p.264 (left column on the left-side page here) about magic in Ephesus:

The Ephesian letters...seem to have consisted of certain combinations of letters or words, which, by being pronounced with certain intonations of voice, were believed to be effectual in expelling diseases, or evil spirits; or which, by being written on parchment and worn, were supposed to operate as amulets, or charms, to guard from evil spirits, or from danger. Thus Plutarch (Sympos. 7) says, "the magicians compel those who are possessed with a demon to recite and pronounce the Ephesian letters, in a certain order, by themselves."

(screen shot of this segment is at the end of this posting)

There is no reason to doubt that "miracles" such as "expelling demons" could be done through these "Ephesian letters" as was stated by Plutarch. We already know from the case of Simon (referred to above) that people were misled thinking that his powers came from God, so apparently those abilities were real.

Demons would likely had wanted to give the impression that such incantations work at least some times, in order to hook people in believing in them.

This is all not particularly surprising in light of 2 Corinthians 11:14-15:

14 And no wonder, for Satan himself masquerades as an angel of light. 15 It is not surprising, then, if his servants also masquerade as servants of righteousness. Their end will be what their actions deserve.

The reason for this is that what matters to Satan is that we do not find the path to serve God the right way. Other than that he does not care who or what our God is; it could be money (Matthew 6:24) or our own gut (Philippians 3:19). The issue of whom we worship matters as can be seen in the 3 tests where he was trying to get Jesus to break God's laws (Matthew 4:1-10) in seemingly small ways (not through murder, adultery, or such things). The only test where Satan actually offered something to Jesus in return was the last one in which he offered to give Jesus all earth's kingdoms for an act of worship.

Here is part of the p.264 from the book "Notes, Explanatory and Practical, on the Acts of the Apostles" by A. Barnes: