I hate answering my own question, but @Matt Bodily put me on the right track.
The @Html.Action method actually invokes a controller and renders the view, so that wouldn't work to create a snippet of HTML in my case, as this was causing a recursive function call resulting in a StackOverflowException. The @Url.Action(action, controller, { area = "abc" }) does indeed return the URL, but I finally discovered an overload of Html.ActionLink that provided a better solution for my case:
@Html.ActionLink("Admin", "Index", "Home", new { area = "Admin" }, null)
Note: , null is significant in this case, to match the right signature.
Documentation: @Html.ActionLink (LinkExtensions.ActionLink)
Documentation for this particular overload:
LinkExtensions.ActionLink(Controller, Action, Text, RouteArgs, HtmlAttributes)
It's been difficult to find documentation for these helpers. I tend to search for "Html.ActionLink" when I probably should have searched for "LinkExtensions.ActionLink", if that helps anyone in the future.
Still marking Matt's response as the answer.
Edit: Found yet another HTML helper to solve this:
@Html.RouteLink("Admin", new { action = "Index", controller = "Home", area = "Admin" })