With chngcntr it's easy:
\documentclass[10pt,a4paper]{article}
\usepackage{chngcntr}
\counterwithin*{section}{part}
\author{wsy}
\title{test}
\begin{document}
\maketitle
\tableofcontents
\part{part 1}
\section{part 1, section 1}
\part{part 2}
\section{how to make this section label become 1}
\end{document}
Note \counterwithin*, so the definition of \thesection is not changed.

If you want arabic numbers for parts and sections in the style part.section, you can do like
\documentclass[10pt,a4paper]{article}
\usepackage{chngcntr}
\usepackage{xpatch}
\renewcommand{\thepart}{\arabic{part}}
\counterwithin{section}{part}
\makeatletter
\xpatchcmd{\l@section}{1.5em}{2em}{}{}
\xpatchcmd{\l@subsection}{1.5em}{2em}{}{}
\xpatchcmd{\l@subsection}{2.3em}{2.8em}{}{}
\makeatother
\author{wsy}
\title{test}
\begin{document}
\maketitle
\tableofcontents
\part{part 1}
\section{part 1, section 1}
\subsection{what's this}
\part{part 2}
\section{how to make this section label become 1}
\end{document}
Similar modifications are needed if you also have subsubsections in the table of contents.

articledoes not provide chapters. Are you looking forreportclass? – Johannes_B Dec 03 '16 at 07:45\usepackage{chngcntr} \counterwithin{section}{part}– Johannes_B Dec 03 '16 at 07:50