I would like offer the opposite suggestion as Brian Wall, which is to
find places where functionality is MISSING or BROKEN in the work place
and your work flow, and provide a FOSS solution for it.  That will open
doors quicker than trying to replace something that your management
views as "working".  Case in point, my first real use for Linux was to
replace a buggy Checkpoint firewall (from the late 90's).  Once I
managed to do that, another project rolled in to host the company
website on Apache.  That was followed by setting up a VPN between sites
using our firewall.  That in turn was followed by...

You get the picture.  When you find a problem that is unsolveable or too
expensive to fix in the proprietary software world, propose FOSS.  Once
they get used to having it around, it'll be easier to convince them to
use it to replace WORKING proprietary solutions.  This is the big hurdle
you must jump.

One thing you must not do is get blinded by the idea that FOSS is the
only solution.  If you don't view things pragmatically from a business
perspective or your managers' POV's, you're not going to get anywhere.
"When you have a hammer, everything looks like a nail," just doesn't
work.

Chad