Is e-mail from Microsoft headquarters saying I won a prize legit?

No. These e-mails are scams, spam, or phishing e-mails. They're intended to trick you into forwarding the e-mail to your friends or family in hopes of getting a prize, or donate money to a worthy cause. They're sometimes used as well to get personal information from you such as your name, e-mail, phone, etc.

These e-mails are not sent by a Microsoft promotion team or any other part of Microsoft and our recommendation is to delete the message.