With risks growing that Ebola could flare on foreign shores, the U.S. is calling for nations to dispatch doctors and nurses to West Africa, where thousands of lives are on the line. Few have heeded the call, but one country has responded in strength: Cuba.

In the weeks since U.S. President Barack Obama sent the first of nearly 4,000 troops to West Africa, the struggle to quell Ebola has created odd bedfellows. Perhaps none is quite so odd as the sight of Cuban doctors joining forces with the U.S. military to combat Ebola in...