EDIT 23/08/2016
I presume you are in cookie authentication with identity
// Enable the application to use a cookie to store information for the signed in user
// and to use a cookie to temporarily store information about a user logging in with a third party login provider
// Configure the sign in cookie
app.UseCookieAuthentication(new CookieAuthenticationOptions
{
AuthenticationType = DefaultAuthenticationTypes.ApplicationCookie,
LoginPath = new PathString("/Account/Login"),
Provider = new CookieAuthenticationProvider
{
// Enables the application to validate the security stamp when the user logs in.
// This is a security feature which is used when you change a password or add an external login to your account.
OnValidateIdentity = SecurityStampValidator.OnValidateIdentity<ApplicationUserManager, ApplicationUser>(
validateInterval: TimeSpan.FromMinutes(30),
regenerateIdentity: (manager, user) => user.GenerateUserIdentityAsync(manager))
}
});
This is the default configuration with identity in Visual Studio.
I can argue why it is not a good option for security but that's not the point.
You can go whit it in "postman" but it's tricky
this is how I do it :
- Make a request over your login page :

- Get the anti forgery token in the form :

- Make a post request on login page with this post params in data form :

Now your postman get the authentication cookie and you can request web api with [authorize] tag
EDIT
For tool you have to add an authorization header.
- Go in the Headers form
- Add the HTTP header "authorization"
- Click on the edit button et voilà ;)
screen shot
Previous answer deleted