Acts 19:13 informs us about the existence of iterant Jewish exorcists who, inspired by Paul, decided to undertake the practice of invoking Jesus' name to cast out demons, probably because they wanted to reach the same success ratio as Paul:
13 Then some of the itinerant Jewish exorcists undertook to invoke the name of the Lord Jesus over those who had evil spirits, saying, “I adjure you by the Jesus whom Paul proclaims.” [Acts 19:13, ESV]
Based on later verses it appears that they managed to see some success and the practice gained some traction, until they came across an evil spirit that was way too strong for their level. All this is very interesting and food for thought, but for this question I would like to bring your focus to the fact that there were Jewish exorcists in the first place. We know this because Acts 19:13 says "[...] some of the itinerant Jewish exorcists undertook [...]". This means that there were many itinerant Jewish exorcists around at the time and only a fraction of them began to invoke the name of Jesus (meaning that the rest continued doing their exorcisms the way they used to do them).
Question: How did the itinerant Jewish exorcists cast out demons before they undertook to invoke the name of Jesus?
Evidently they couldn't have used the name of Jesus to cast out demons before they started using the name of Jesus (duh), so they had to have employed a different method and receive authority from a different source prior to that. In fact, a more generalized version of the question would be: How did Jewish exorcists (and people in general) cast out demons before the name of Jesus became popular? I don't know if these are questions that can be answered relying solely on the Bible, so I'm open to answers that incorporate useful insights from historical records.
A closely related question for the interested reader: How do Christians make sense of exorcisms in other religions?