If you're adding ASP.NET WebApi inside ASP.NET MVC you probably want to respond unauthorized instead of 302 requests. As you know the ASP.NET infrastructure come into play and when you try to set response status code to HttpStatusCode.Unauthorized you will get 302 code, then redirect to login page.

Sometime you need to return a 401 unauthorized instead of 302, you can use the code below to solve that issue

First you need to open the Startup class, then add IsApiRequest method to help you check the Web Api request

private static bool IsApiRequest(IOwinRequest request)
{
    return request.Uri.LocalPath.StartsWith(VirtualPathUtility.ToAbsolute("~/api/"));
}

Update ConfigureAuth method as below

public void ConfigureAuth(IAppBuilder app)
{
    // Configure the db context, user manager and signin manager to use a single instance per request
    app.CreatePerOwinContext(ApplicationDbContext.Create);
    app.CreatePerOwinContext<ApplicationUserManager>(ApplicationUserManager.Create);
    app.CreatePerOwinContext<ApplicationSignInManager>(ApplicationSignInManager.Create);

    // 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)),
            OnApplyRedirect = ctx =>
            {
                if (!IsApiRequest(ctx.Request))
                {
                    ctx.Response.Redirect(ctx.RedirectUri);
                }
            }
        }
    });
    app.UseExternalSignInCookie(DefaultAuthenticationTypes.ExternalCookie);

    // Enables the application to temporarily store user information when they are verifying the second factor in the two-factor authentication process.
    app.UseTwoFactorSignInCookie(DefaultAuthenticationTypes.TwoFactorCookie, TimeSpan.FromMinutes(5));

    // Enables the application to remember the second login verification factor such as phone or email.
    // Once you check this option, your second step of verification during the login process will be remembered on the device where you logged in from.
    // This is similar to the RememberMe option when you log in.
    app.UseTwoFactorRememberBrowserCookie(DefaultAuthenticationTypes.TwoFactorRememberBrowserCookie);

    // Uncomment the following lines to enable logging in with third party login providers
    //app.UseMicrosoftAccountAuthentication(
    //    clientId: "",
    //    clientSecret: "");

    //app.UseTwitterAuthentication(
    //   consumerKey: "",
    //   consumerSecret: "");

    //app.UseFacebookAuthentication(
    //   appId: "",
    //   appSecret: "");

    //app.UseGoogleAuthentication(new GoogleOAuth2AuthenticationOptions()
    //{
    //    ClientId = "",
    //    ClientSecret = ""
    //});
}

Now you can test your Web Api

[Authorize]
[HttpGet]
public async Task<object> Get()
{
    return "Protected area";
}

I hope so you can solve the problem