\documentclass{article}

\usepackage[numbers]{natbib}

\usepackage{notoccite}

% If you have \cite commands in \section-like commands, or in \caption, the

% citation will also appear in the table of contents, or list of whatever.

% If you are also using an unsrt-like bibliography style, these citations

% will come at the very start of the bibliography, which is confusing. This

% package suppresses the effect.

​

\begin{filecontents}{\jobname.bib}

@misc{foo,

 author = {Foo},

 title = {Title of Foo},

}

@misc{bar,

 author = {Bar},

 title = {Title of Bar},

}

@misc{baz,

 author = {Baz},

 title = {Title of Baz},

}

\end{filecontents}

​

\begin{document}

\listoffigures

​

%\chapter{Introduction}

\section{The Beginning}

Here we cite \citeauthor{foo} \cite{foo}.

Now we cite \citeauthor{bar} \cite{bar}.

Finally, we cite \citeauthor{baz} \cite{baz}.

\begin{figure}[htbp]

\centering

\framebox[0.5\textwidth]{Here's a figure.}

\caption{Figure from \cite{bar}, which should most definitely \textbf{not}

         be numbered [1] in the LoF}

\end{figure}

\bibliographystyle{unsrtnat}

\bibliography{\jobname}

\end{document}