  Given a compact K\"ahler manifold $X$, there is an equivalence of categories
between the completely reducible flat vector bundles on $X$ and the polystable
Higgs bundles $(E,\, \theta)$ on $X$ with $c_1(E)= 0= c_2(E)$ \cite{SimC},
\cite{Cor}, \cite{UY}, \cite{DonI}. We extend this equivalence of categories to
the context of compact Sasakian manifolds. We prove that on a compact Sasakian
manifold, there is an equivalence between the category of semi-simple flat
bundles on it and the category of polystable basic Higgs bundles on it with
trivial first and second basic Chern classes. We also prove that any stable
basic Higgs bundle over a compact Sasakian manifold admits a basic Hermitian
metric that satisfies the Yang--Mills--Higgs equation.
