documentclass[12pt, letterpaper]{article}
usepackage[colorlinks=true]{hyperref}

begin{document}

Here is a sentence that refers to Refs.~cite{ref1,ref2,ref3}.

I would also like to refer to these references as [1--3].

The closest I can get is cite{ref1}--cite{ref3}.

begin{thebibliography}{5pt}
  bibitem{ref1}
    J.~Doe, Some title 1.
  bibitem{ref2}
    J.~Doe, Some title 2.
  bibitem{ref3}
    J.~Doe, Some title 3.
end{thebibliography}

end{document}