I suggest you (a) use subfigure environments instead of minipage environments, (b) assign zero width to the subfigure environments, and -- most importantly -- (c) use \refstepcounter directives instead of caption directives to increment the counter named subfigure without generating any text. The \label directives should remain in place, of course.
Doing so will print out the overall caption below the composite graph and generate the following sentence:
Figure 1 has 1a on the left and 1b on the right, also known as a and b.
In case you're wondering what's going on here: LaTeX's \caption directive not only typesets its argument as the caption text (duh), it also invokes a \refstepcounter instruction to increment the counter (called subfigure) associated with the current float. It's this counter that the \label instruction needs to find and latch on to.
It's also worth mentioning that the subcaption package actually provides a dedicated macro called \phantomsubcaption for exactly your use case, viz., not wanting to produce a subcaption-related text while preserving the ability to create cross-references to the subfloat(s). \phantomsubcaption (essentially) performs the instruction \refstepcounter{subfigure} for you. Use \phantomsubcaption in lieu of \refstepcounter{subfigure} if you prefer. In fact, if you wish to use \phantomsubcaption, it's not actually necessary to set up zero-width subfigure environments; just enclose the \phantomsubcaption and \label statements in a TeX group, like so:
{\phantomsubcaption\label{left} \phantomsubcaption\label{right}}
A full MWE:
\documentclass{article}
\usepackage{graphicx,subcaption}
\begin{document}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{example-image-a}%
% We now insert 2 zero-width 'subfigure' environments
% to the **right** of the graph. By encasing the
% '\refstepcounter{...}\label{...}' statements inside
% 'subfigure' environments, the subsequent '\subref'
% instructions will generate the expected output.
\begin{subfigure}{0\textwidth}
\refstepcounter{subfigure}\label{left}
\end{subfigure}%
\begin{subfigure}{0\textwidth}
\refstepcounter{subfigure}\label{right}
\end{subfigure}
\caption{Overall caption}\label{figure}
\end{figure}
Figure~\ref{figure} has \ref{left} on the left and \ref{right} on the right,
also known as \subref{left} and \subref{right}.
\end{document}