A: It's stealth, not invisibility. They are two different mechanics.
B: It does break stealth up to TBC when it changed. That is why there is a talent "improved sap" which reduces the chance of sap breaking stealth.
"Sap no longer removes you from stealth when used."
https://web.archive.org/web/20101204052 ... h2p10.html"World of Warcraft Client Patch 2.1.0 (05/22/2007)" under rogues, or just do a search for "sap" on that page.