Either article could be considered correct, depending on the circumstances and the assumptions involved.
On one hand, we can use the indefinite article, because we are merely talking about some email that has been sent. As Michael Rybkin mentioned in an earlier answer, sentences like these are perfectly grammatical and frequently heard in everyday speech:
- I took an apple out of the fruit bowl.
- I bought a washing machine last week.
- I sent a package to my uncle.
- I sent an email to Mike two hours ago.
However, you're onto something when you mentioned that the site knows which email is being referred to. There are many possible situations where using the definite article would also be acceptable. One very simple one, often quoted in elementary grammar books, is that the item being alluded to has been previously mentioned.
Here is an apple. I took the apple out of the fruit bowl.
But this "previous mention" need not always be so obvious, explicit or immediate. For example, consider these snippets of dialogue between spouses:
Q: Hey, why is there only $200 left in our checking account?
A: Remember? I bought the washing machine last week.
Q: I didn't hear you leave. Where did you go?
A: To the post office; I sent the package to my uncle.
The first answer sounds fine, because we assume there is only one washing machine in the house. The second answer would be fine, too, if we can assume the couple may have had a dialogue a few days ago that went something like this:
Q: What's up with this big box on the counter?
A: It's a package for my uncle Ernie. His birthday is next week.
As for that last one, we can use the definite article there, too:
Q: Did you send that "save the date" notice to your cousin yet?
A: Yes, I sent the email to Mike two hours ago.
I could say "an email" just as easily as I can say "the email". In either case we know we are talking about an email that contains the "save the date" information.
As for your message, I think this sounds perfectly fine:
We’ve sent an email to email@example.com
However, the company could have also written:
We’ve sent the email to firstname.lastname@example.org
and I'd interpret that as:
We’ve sent the email [that we previously indicated would be sent] to email@example.com
However, we are not forced to use the definite article just because a forthcoming email may have been previously mentioned.
As a footnote, notice how no mention was ever made of the fruit bowl, but the word "the" sounds perfectly natural in that sentence.