Most Emmys Won By a Single Episode: 5
That'd be ER's pilot, "Love's Labor Lost," which took home trophies for writing, directing, editing, and sound editing and mixing (and was nominated for multiple acting awards). Several other episodes have won four Emmys — notably Breaking Bad's stellar "Ozymandias" from its final season, as well as the very first episodes for Lost and Hill Street Blues.
It's a bit of an opposing lesson: Pilots winning awards isn't particularly surprising, given the way they lay the groundwork for a series, represent a specific vision, and often introduce many of a show’s most innovative elements. But "Ozymandias" comes all the way at the end of AMC's groundbreaking drama, playing off all of the tension the show had built up over and the nuances of the actors' performances. It's an indicator of how, as much as we increasingly think about it in terms of shows or seasons, television is still an inherently episodic medium.