The width= setting of the caption package will always center its caption with respect to the \textwidth, so width=1.2\textwidth will give you infact a left and right margin of -0.1\textwidth.
For manipulating its margins, better use margin= instead, since it offers discrete values for left and right margin. For example:
\documentclass{scrartcl} % KOMA-Script article
\usepackage{lipsum}
\usepackage[english]{babel}
\usepackage{graphicx}\graphicspath{Figures}
\usepackage[format=plain,justification=raggedright,font={small}]{caption}
% uncomment this if the setting should affect all figures
%\captionsetup[figure]{margin={0pt,-0.2\textwidth}}
\begin{document}
\lipsum[120]
\begin{figure}[thbp]
\includegraphics[draft,width=1.2\textwidth]{Figures/foo.pdf}%
\captionsetup{margin={0pt,-0.2\textwidth}}% only for this figure
\caption{Caption that has the same width as the figure, but should be aligned to the left as everything else, but should spill out of the right margin}
\label{fig:key}
\end{figure}
\lipsum[120]
\end{document}
As an alternative, one could put the whole figure contents inside a minipage with the desired width, this way both the content and the caption will be adjusted to the new width:
\documentclass{scrartcl} % KOMA-Script article
\usepackage{lipsum}
\usepackage[english]{babel}
\usepackage{graphicx}\graphicspath{Figures}
\usepackage[format=plain,justification=raggedright,font={small}]{caption}
\begin{document}
\lipsum[120]
\begin{figure}[thbp]
\begin{minipage}{1.2\textwidth}
\includegraphics[draft,width=\textwidth]{Figures/foo.pdf}%
\caption{Caption that has the same width as the figure, but should be aligned to the left as everything else, but should spill out of the right margin}
\label{fig:key}
\end{minipage}
\end{figure}
\lipsum[120]
\end{document}
Both should give identical results:
