I'm a native British English speaker and I would have answered : "No I don't!" 

"No I haven't" isn't incorrect but it's not really used by itself in this context unless as part of a longer sentence " No I haven't got any"

You will find "No I haven't" in books more often than in spoken English.

"No I haven't" is what I would reply to questions such as these 

Have you seen the film?

Have you sent the letter?

Have you read the book?

That's pretty interesting.

Your response to the question isn't incorrect, and personally, as a native English speaker, I've never heard "No, I haven't," as a response to that question. Your answer is correct, but the app may be referring to another dialect of English as well. Though within Standard American English, I would not use, "No, I haven't."

The app could be teaching you British English.

Best, x

Haven’t any relates to the question „have you,“ „Do you“ would require an answer with the same verb „Do.“. So „Do you“ is properly answered with „I do“ or „I don’t.“  The app is just simply wrong. 

By the way, I don’t think English has any „dialects“ as such. If by the word you mean the way other languages do. 

Eduardo, I would have written the same answer that you did.   I've lived in the United States all my life and can tell you that people here would usually respond to that question with a simple, "No, I don't."

If would sound slightly odd to me if someone replied to a "Do you have...?" question with just the three words, "No, I haven't."  I might wonder if they were from another English speaking country, such as England or Ireland.  Or I might think that they are trying to be very formal with their speech.  It doesn't sound wrong...just slightly different than the way I usually hear people say it.

Adding additional words after the "No, I haven't" make it sound slightly better (For example, "No, I haven't got any money.").  However, "No, I don't." or "No, I don't have any ____" or "No, I don't have a _____" are definitely my preferred ways to answer a question that begins with, "Do you have..."

I agree with Katarzyna that "No, I haven't." sounds fine in response to a question that begins with "Have you....?"  (Have you done this?  Have you gone there?).