That being said, let's assume the claim is true. I think there's a bit more to the outcome. A quick thought experiment should illustrate closely enough. If it didn't result in unemployment increasing overall, then why not just raise it more? $25, hell, $50 an hour? You intuitively know the answer - most businesses can't afford that, especially when you consider that everyone above those levels currently will also have to be scaled up as well.
So if we know there is a limit to where the minimum can be raised, what does this mean then? It means the underlying claim being made is really "government is smart enough to determine the acceptable mandatory raise in minimum wages that will not harm the economy."
Taking it a bit further, let's take a rational business person who is looking at hiring someone. What's the mental math they do, roughly? They look at how much monetary value the person could generate for the company and then they look at what they will need to pay that person (including benefits, payroll taxes, and all other fringe costs). If the costs are greater than the value, it's a hard no. If they are roughly equal, then it's a maybe since you still have more paperwork and general nuisance stuff to worry about when managing someone so it's a "is the juice worth the squeeze" scenario". And anything solidly in the green is a go.
So, for example, if you own a janitorial service where corporations hire you to come clean at night and you pay your cleaning staff $12 an hour, benefits are $3 an hour and you bill the companies that hire you $18 an hour, you have about $3 per person per hour to account for any fixed overhead for the business and profit. If your state suddenly says you have to pay them $15 an hour, you have to raise your prices. There's no other option unless you let people go and force the remaining to work harder, faster, or do a worse job and rush through.
If, however, your profit margins were much higher and you were charging $20-$25 an hour to the companies, you could simply absorb that difference and you will just be eating the added cost in the form of lower profits if you have a competitive business.
And I think that is where we're really at with this currently. Many of the big companies paying low wages have revenue per employee numbers that far exceed the cost of the employee and as such the increase can be absorbed for those companies. Or, their business is so large and the volume per employee is so much that the increase is negligible. Think Walmart - how much per item would the prices have to be to meet such raises? 3% maybe? I saw a study done in the farming industry that said they could double the low wages of field workers by increasing the cost only 7% because the labor cost of food is about 7%.