My goal is as stated in the question title. There are other questions addressing this, but the answers selected, as best I can tell, don't get along with my other existing code, which is summarized below.
\documentclass[11pt]{article}
\usepackage{amsmath,amsthm,titlesec}
\numberwithin{equation}{section}
\newcommand{\periodafter}[1]{#1.}
\newcommand{\presectionskip}{-1\baselineskip}
\newcommand{\postsectionskip}{0.3\baselineskip}
\makeatletter
\renewcommand{\section}{\@startsection
{section}{1}{0mm}%% name, level, indent
{\presectionskip}% % beforeskip
{\postsectionskip}% % afterskip
{\normalfont\LARGE\bfseries}}% % style
\renewcommand{\subsection}{\@startsection
{subsection}{2}{0mm}
{\presectionskip}
{\postsectionskip}
{\normalfont\Large\bfseries}}
\renewcommand{\subsubsection}{\@startsection
{subsubsection}{3}{0mm}
{\presectionskip}
{\postsectionskip}
{\normalfont\normalsize\bfseries}}
\makeatother
%\usepackage[font=bf]{caption}
%\titleformat{\section}{}{\thesection.}{1ex}{\periodafter}
%\titleformat{\subsection}{}{\thesubsection.}{1ex}{\periodafter}
%\titlelabel{\thetitle.\quad}
\begin{document}
\section{Good section}\label{sec:refl}
\subsection{Superior subsection}\label{tableworks}
\subsubsection{This, the best of all subsections}
\end{document}
Thanks for any ideas on how to get this to work.

egreganswer, but obviously failed to try the second. :/ But thank you so much! What do you mean about references? – jdc Dec 15 '14 at 08:02\refor similar. Makes sense to have it like this, in your case. But you have to consider that "chapter 1. is showing..." could be a bit difficult to read especially, if the dot is set at the end of a line. – LaRiFaRi Dec 15 '14 at 09:47\Crefs. I tried a\refas well to check, and there was no dot. – jdc Dec 15 '14 at 10:04titlesecfor all this stuff. – LaRiFaRi Dec 15 '14 at 10:09titlesec, although I don't remember what I'm using it for. That's the danger with understanding as little as I do and adapting advice one finds online, I suppose. – jdc Dec 15 '14 at 10:15