MATLAB is a Java interpreter. See Java Libraries in MATLAB help. Now it just so happens that there is an excellent Java implementation of Git, the most widely used distributed version control system, called JGit. So it seems obvious that the best way to use Git in MATLAB is with JGit, and it turns out this is surprisingly easy, because JGit has a porcelain class.
I have written a ridiculous wrapper to use JGit in MATLAB that you can clone from Github or download from the MATLAB Central File Exchange. Try my wrapper, but then either contribute or branch out and just use the JGit package. It is surprisingly easy.
NOTE: Did I mention that you do not need to install anything! JGit is your Git client. JGit is not a wrapper for C/C++/Perl scripts. It is a pure Java implementation of Git. That really makes this the one of the easiest way to use Git if you are also using MATLAB.