You're right!

Yes, the reasoning is fallacious. It is another instance of the gambler's fallacy, since the winning numbers today are entirely independent of the winning numbers yesterday. This version of the gambler's fallacy is very tempting, as it is true that the probability of the same numbers coming up two days in a row is very small--about one in 10 million. So the chance of winning if you pick yesterday's numbers is only one in 10 million. But notice that the chance of winning is the same whatever numbers you choose; for any set of six numbers you choose, the chance of getting them all correct is one in 10 million.

In fact, an argument can be made that you are better off picking yesterday's numbers. That's because if two people win, they have to share the prize. Since people tend to avoid yesterday's numbers (as well as "unlucky" numbers like 13), if you choose yesterday's numbers you are less likely to have to share your prize!