Would getting a faster MSc instead of just BSc affect admission chances for PhD? I will try to describe my situation briefly.
The university I'm currently studying at offers a 4-year Bachelor program in mathematics.

I am currently in my 3rd year and the program for the 4th year consists only of optional courses (which I choose). As I have taken a rather high amount of optional courses so far, I can satisfy my ECTS requirements for the whole program in 3 years. The university, in this situation, would allow me to graduate at the end of my third year. Then, I could pursue an MSc. degree at the same university for a year and graduate with an MSc at the end of my 4th year of study in the university.

I would like to note that this has been done before by other students here and I'm not fantasizing.

My intention is to apply for a PhD at a university in the USA after the 4th year. (I am not a US citizen and I haven't studied there).

My question is: would getting an earlier BSc and MSc degree affect (negatively or positively) my chances for admission in a top university in the USA?
Should I just graduate with a BSc. in math?
I know that sometimes having a MSc. from the same university can be a negative sign. Should I be worried about that?

Some other background:
My current university is not really "elite" of any sort; it barely enters the top 600 in the world. I have a few published articles and I have participated in conferences. I am also currently pursuing an internship in a research institute in the country.

Thank you.
(I'm on the admissions committee of a top-10 US computer science department; my advice may be way off base for mathematics PhD programs.)

All else being equal, getting a master's degree in one year would probably count against you.  You're far better off taking some graduate-level courses, and possibly getting some research experience, while staying in the BSc progam.

At least in my department, PhD applicants who already have master's degrees are held to a significantly higher standard than applicants who only have bachelor's degrees.  While having formal research experience is an advantage for undergraduate applicants, it's a de facto requirement for applicants with an master's degree.  One year is not a lot of time to get some real (meaning publishable) research experience, especially since most one-year master's programs have heavy coursework requirements.  And you'll be competing with other MS applicants who've already spent two years in graduate school.

My department steers PhD-oriented undergraduates away from our 5-year BS/MS program for this precise reason.