Firstly, you cannot assume that it was the questioner who issued the downvote. It could be anyone.
If you feel your answer is incorrect but have a better one, you have a number of options as to what to do about it, primarily:
1) Do nothing. Just live with it and move on. Everybody gets downvotes from time to time. It is not impossible that you will get other votes, up and down.
2) Delete the answer. You feel that the answer doesn't match the question, so you feel that it may not be appropriate to leave it there. If you do this within a couple of days, you will get the two points of reputation back. Bear in mind that the post may actually get flagged Not an answer by enough people to delete it anyway.
3) Edit the answer. This is fine. You put up a new answer with which you feel happy and the site as a whole benefits from it. You may well get upvotes. The answer that you don't like is removed.
There are also combinations:
4) Delete the answer, then edit it, then undelete it.
As the original poster of the answer, you will be able to see it even if deleted. This way you can edit it in your own time without worrying if someone else will downvote it while you work.
5) Do nothing with the first answer, just post a new answer in a new post.
6) Delete the answer, then post a new answer in a new post.
This has the benefit of giving you the reputation back on the old answer and the new answer starts from zero.
You are trying to improve the quality of the whole Q&A post. This is appreciated.
Note that any deleted answer places a mark against your account. A note that you have had an answer deleted (whether deleted by you or by others). There is no need to worry about this - action is only taken if you wind up with several consecutive answers deleted. In that case the SE system may block you from answering for a while - this is an automatic action.