Q:

How could the person in Luke 9:49 cast out demons without Jesus' mandate?

Luke 9:49 (ESV):

John answered, “Master, we saw someone casting out demons in your name, and we tried to stop him, because he does not follow with us.”

The 12 (and later the 70) only started healing people and casting out demons AFTER Jesus "gave them power and authority over all devils, and to cure diseases."
The Holy Spirit had not been sent yet, so with what authority was this person acting? 
I am not questioning the good will of the anonymous healer. Was he an imposter Jesus would have stated so.
What I'm really wondering is that people didn't need a direct command (or mandate) from Jesus to act in His name?  

A:

DEV--Don's Edited Version
The answer to your question comes from Jesus, and also from the words of John in the verse you quoted. 

John's words: John assumed that because the unnamed exorcist was not in the inner circle of the Twelve that his exorcism could not therefore be on the "up and up." 
Jesus' words:  In essence, Jesus told John (and all the other disciples who were present during the "Who is the greatest?" incident), 

"Hey [he didn't say 'Hey'], if that person you speak of exorcised demons in my name successfully, then he must be one of us, even though he is not a member of this particular group of twelve disciples."

I think many Christians (and I count myself among that number) assume that the only disciples who performed miracles in Jesus' name were those whom Jesus sent out (or commissioned, as in Mark 6:7, Luke 9:2 and 10:1) to precede him and pave the way for him by preaching the kingdom and by performing miracles such as exorcisms and perhaps other works of healing. This assumption may not be warranted, particularly in light of Jesus' words; namely, 

"'Do not hinder him [i.e., the man who cast out demons], for he who is not against you is for you'" (Luke 9:50)

In other words, Jesus could very well have commissioned this man on a one-to-one basis. Perhaps for whatever reason this man could not "follow along with"--as John put it--the Twelve or some other recognizable group of Jesus-followers (e.g., "the 70" in Luke 10:1), so Jesus permitted him to heal people in Jesus' name. 
I think it unwise to assume, for example, that this man was a nonbeliever (similar to Simon Magus in Acts 8:9 ff.) and a "Lone Ranger" who was performing miracles without Jesus' sanction. Jesus could very well have sanctioned him to perform these miracles. Moreover, as I've suggested above, Jesus seems to have assumed that since the man performed miracles "in Jesus' name," that he was indeed "one of us"--meaning the Twelve and all his other disciples, even if John did not recognize him as such.
In conclusion, there is no reason of which I am aware why this "lone wolf exorcist" was any more in need of the sending authority of the Holy Spirit than were the Twelve. After all, the Holy Spirit did not commission the Twelve (and others); Jesus did. On the authority of Jesus, the Twelve and this unnamed disciple were given the ability to heal. The power to do so undoubtedly came from the Holy Spirit, but the disciples were not even aware at this point that there was such a person called the "Holy Spirit." That awareness came later (e.g., in John 15). 
Additional Thought/Addendum
Jesus had a way of surprising--shocking even--his closest followers and disciples. Example par excellence: Jesus' conversation with the Samaritan woman at the well of Sychar, in John 4:4-45. This woman had a number of strikes against her, at least from the disciples' perspective:

She was a she. In other words, she was a second- or third-class citizen, as were most women in those days (with a few exceptions, to be sure).
She was a member of a despised people group, the Samaritans, whom the Jews considered to be a bunch of religious half-breeds--which was partly true, since after the divided kingdom and the subsequent exile of Israeli Jews, Samaria was "seeded" with the importation of foreign colonists who brought their own syncretistic views of religion to Palestine. Consider Wayne Brindle's comments:

"The development of Samaritanism and its alienation from Judaism was a process that began with the division of the kingdom of Israel, and continued through successive incidents which promoted antagonism, including the importation of foreign colonists into Samaria by Assyria, the rejection of the new Samaritan community by the Jews, the building of a rival temple on Mt. Gerizim, the political and religious opportunism of the Samaritans, and the destruction of both the Samaritan temple and their capital of Shechem by John Hyrcanus during the second century B:C. The Samaritan religion at
  the time of Jesus had become Mosaic and quasi-Sadducean, but strongly anti-Jewish. Jesus recognized their heathen origins and the falsity of their religious claims."

She was a woman of questionable morals. (Today, a bigot would probably label her a slut!)
She was alone with Jesus and conversed with him in a give-and-take discussion, which could have lasted an hour or more. Pious Jewish men simply did not talk with women in public, let alone have a religious confabulation. Unthinkable!
She gave Jesus a drink of water from her own bucket, which to the Jews of Jesus' day would be unthinkable (just think of the ritual uncleanness of it all!). 

Nevertheless, Jesus, John tells us, "had to pass through Samaria" (4:4). Why? Because a Samaritan woman had a divine appointment with her soon-to-be Savior and Lord! Jesus could have bypassed Samaria as he travelled from northern Israel to southern Israel, as did his fellow Jews, but he was never one to depart from the path his Father had chosen for him. Furthermore, his Father and he shared a love for outcasts, outsiders, the foreigner, the stranger, and the alien. Why else would the Law of Moses contain so many instructions regarding the humane treatment of non-Hebrews with whom the Israelis interacted.
My point is this: John--and perhaps the other disciples--did not realize at the time he registered his opinion about the "interloper" who was casting out demons in Jesus' name 
that Jesus' compassionate heart had room for outsiders and interlopers, such as the Samaritan woman at the well, and also the Syro-Phoenician woman whose daughter was plagued by a demon:

"Then Jesus went thence, and departed into the coasts of Tyre and Sidon. And, behold , a woman of Canaan came out of the same coasts, and cried unto him, saying , 'Have mercy on me, O Lord, thou Son of David; my daughter is grievously vexed with a devil.' But he answered her not a word. And his disciples came and besought him, saying , 'Send her away; for she crieth after us.' But he answered and said, 'I am not sent but unto the lost sheep of the house of Israel.' Then came she and worshipped him, saying, 'Lord, help me.' But he answered and said, 'It is not meet to take the children's bread, and to cast it to dogs.' And she said, 'Truth, Lord: yet the dogs eat of the crumbs which fall from their masters' table.' Then Jesus answered and said unto her, 'O woman, great is thy faith: be it unto thee even as thou wilt.' And her daughter was made whole from that very hour'" (Matthew 15:21-28 KJV).  

In conclusion, the people to whom Jesus reached out may not have fit the mold of "disciple material" which was in John's (and others') mind, but then Jesus was not big on molds. Jesus was more interested in people's hearts, and he was an equal opportunity Savior, Healer, and Lord, and not just to "the lost sheep of the house of Israel."  

