RH Webster: the deepest feelings; the strongest affections: to tug at one's heartstrings. [..orig. alluding to tendons that were thought to brace the heart]
Of course it exists and I am surprised that no one was familiar with it. Perhaps the expression is somewhat old-fashioned... e.g., You would use it if you heard a particularly touching story, one that could almost bring you to tears: "When I heard of her losses, it really tugged at my heartstrings."

Der Kleine Muret-Sanders

Certainly English speakers should have known this word. However, it exists almost exclusively in the context RES gave, that is, "tug[ged] at my heartstrings" (sometimes "pulled").

You'll have a hard time finding it anywhere else in plain, running English text. If you do a careful search, restricting results to English, including "heartstrings" but excluding forms of "tug", you still have around 40,000 hits, but they are all names of books, websites, stores, newsletters, song titles, and so on. Try to find a simple sentence outside the given context and it's very hard.

I would thus consider it a cliche, and requires you to translate the whole phrase, and not just the word by itself which is almost meaningless when it stands alone.

In Harry Potter, dragon heartstring means "Herzfaser vom Drachen"