Question: When I budget for me, I usually count savings part of the expenses.

**Is it the correct way to do it?** Because of this, when I look at my budget at the end of the month, it feels like my expenses are a bit high.

Answer: 
It's a matter of semantics. 

If a budget is done to include every last item, by definition there would be no extra. I save about 20% of my income, but if that's in my budget, the budget is a 100% view of my gross income, a pie chart (or table) of all cash flow. 

Some people prefer to think of their budget as the spending with what remains going into saving. In the end, that produces the same result. An accounting of all of your money, although this approach strikes me as having the potential of the savings aspect appearing an afterthought. 

Most financial advisors would council a client to save "off the top" and not with what remains after everything else. 

From looking at some of the other answers I see the issue is one of accounting. If it helps the matter, the piece you are missing in the discussion is the rest of the balance sheet, the assets accruing. If you gross $2000, account for $2000, but $400 went to savings, your balance sheet has improved $400.

By the way, paying a debt's principal will also improve the balance sheet. Your net worth grows for saving a dollar or paying off a dollar's worth of debt.
Savings isn't really an expense but I understand what your trying to say, you are allocating a specific amount to save. Which is definitely a good idea. If you're going over the budget then you have to decide what to cut down on. You can sacrifice some luxuries or if you feel comfortable enough you can cut back on how much you save. Clearly the more conservative choice would be to cut back on luxuries.