Sanskrit and most European languages do share a common ancestor. But Sanskrit is not that common ancestor, and the influence of Sanskrit itself on European languages is trivial, mostly limited to words relating to Indian religions, yoga, and so on.

I think this confusion arose because it was similarities between Greek, Latin and Sanskrit that first alerted scholars to the existence of a common ancestor.

You can see a family tree of the Indo-European languages at https://upload.wikimedia.org/wikipedia/commons/4/4f/IndoEuro...