Looking at Translate() from LanguageManager -> it uses current UI culture (which is in your case is "sv").

return this.Translate(key, Thread.CurrentThread.CurrentUICulture.IetfLanguageTag);

Try to "restore" desired culture just before calling Translate().

AFAIK EPiServer content language really does not influence resource translation language.

EPiServer's current language context is lost when posting using ajax. Here is how you can set language context inside a web api call:

var language = ControllerContext.RouteData.Values["language"].ToString();
Thread.CurrentThread.CurrentUICulture = new CultureInfo(language);
Thread.CurrentThread.CurrentCulture = new CultureInfo(language);
            
EPiServer.BaseLibrary.Context.Current["EPiServer:ContentLanguage"] = new CultureInfo(language);

Thank you all for the help. I have overridden InitializeCulture() method in the base class(for ajax pages) and used following code:

protected override void InitializeCulture()
{
    base.InitializeCulture();

    Thread.CurrentThread.CurrentUICulture = EPiServer.Globalization.ContentLanguage.PreferredCulture;
    Thread.CurrentThread.CurrentCulture = EPiServer.Globalization.ContentLanguage.PreferredCulture;
}

Property 

EPiServer.BaseLibrary.Context.Current["EPiServer:ContentLanguage"]

     contain the correct culture even without any changes.

Running into the same issue. Where do you override InitializeCulture()?

If you are using WebForms, it's an override of PageBase.

If you are using MVC, take a look at http://stackoverflow.com/questions/21160599/custom-language-handling-episerver