How to send steam gift game to the winner?

You can add the winner on Steam and send him the gift, but you will also see the winners email, and can send the gift through there.

After the GA ended you can send gift via email and then click the send button

Pick any option:

1. Send it to his email (you can find it by clicking the Winners button in the left sidebar of your giveaway)
2. Add him to your friend list on Steam and send it to him directly
3. Add him to your friend list on Steam and use the trade window to send the game
4. Generate the gift link by sending the game to your own email, then copy-paste the gift link into the Key field inside your giveaway to deliver it via the site using the Send button

Clicking the Send button without storing the key or gift url will simply indicate that you used some other delivery method and will get rid of the notification in the header.