Because the oil price decrease, I can save some money for family expense. sembra naturale?

It would sound more natural if the verbs (doing words) had the "-ed" at the ending. It would sound more natural if you put:

"Because OF the oil price THAT decreased, I can save some money for family expense" however you can put "MY family expense" if you want to be even more natural.

If you want the decrease to be continuous, then it would be more natural for you to say:

"Because the oil price IS decreasing"

You could also say:

"Because of the oil price decrease"

This example would fit yours better however.

So...
"Because of the oil price decrease, I can save some money for my family expense."