You don't want unnumbered sections, you just want to modify the style not to show the number:
\documentclass{article}
\makeatletter
\def\@seccntformat#1{%
\expandafter\ifx\csname c@#1\endcsname\c@section\else
\csname the#1\endcsname\quad
\fi}
\makeatother
\begin{document}
\section{Section aa}
\subsection{Subsection}
\subsection{Subsection bbb}
\section{Section zzz}
\subsection{Subsection jjj}
\end{document}
Or as clarified in comments you do want the number, but prefixed with Section
:
\documentclass{article}
\makeatletter
\def\@seccntformat#1{%
\expandafter\ifx\csname c@#1\endcsname\c@section
Section \thesection:
\else
\csname the#1\endcsname\quad
\fi}
\makeatother
\begin{document}
\section{aa}
\subsection{Subsection}
\subsection{Subsection bbb}
\section{zzz}
\subsection{Subsection jjj}
\end{document}
\section{hello!}
and just modify the style to putSection 1:
at the left instead of just1.