Hi Tomasz

If your API controller inherits from UmbracoApiController (which in turn inherits UmbracoApiControllerBase)

https://github.com/umbraco/Umbraco-CMS/blob/9c06ef83e217affbce51c1d6682f10914be2b400/src/Umbraco.Web/WebApi/UmbracoApiControllerBase.cs

You'll have access to the PublishedContentCache through the UmbracoContext property of the controller, this in turn has a 'Content' property that is of type IPublishedContentCache which is where the GetByRoute() method lives:

 public class SuperApiController : UmbracoApiController
    {
        [System.Web.Http.HttpGet]
        public IEnumerable<string> GetListOfThings()
        {
            // here is GetByRoute...
            UmbracoContext.Content.GetByRoute();

        }
    }

regards

Marc