Unfortunately, that’s really what effectively needs to be done, when using the edit-then-launch workflow. You could try to automate the process (write a PythonScript which would save the current file, grab the filename, then search through other open documents, looking for a HTML file that links to that css filename, then launches that HTML). But, honestly, that seems like the hard way to me.
If it were my workflow, when I opened the HTML and the CSS, I would load the HTML into my browser. Then any time I save the HTML or CSS file, I would just hit the browser’s reload button to refresh what’s there.