From my point of view, your translation is very correct. The expression is commonly used in French, and I would even say that it is a very popular one.

De mon point de vue, votre traduction est tout à fait correcte. L’expression est couramment utilisée en français, et je dirai même que c’est une expression très populaire.

L’avenir appartient à ceux qui se lèvent tôt is certainly the most pertinent French equivalent to the early bird catches the worm.

It conveys a very similar meaning and tone, even though it is far from a literal translation.

I’m French I confirm this is perfectly fine.
The idea behind it is that people waking earlier would have more time to work (and reach success). It was at a time agriculture was predominant and working earlier meant being more productive.
Regards.