http://revach.net/article.php?id=4752

Section: Hanhagos   Category: Minhagim
Why Do We Say Selichos At Night On Motza'ei Shabbos?
The Shulchan Aruch (581:1) says that the Minhag is to say Slichos early in the morning.  The Mogen Avrohom says that the ned of the night right before the morning is a special Eis Ratzon.  Others are noheg to say at night but not before Chatzos.  Some say that this time is less preferable.  

If so, why on the first night of Slichos do we start Motza'ei Shabbos at Chatzos, why not wait until the morning?  One of the reasons is that the Piyut starts B'Motza'ei Yom Minucha, which refers to Motza'ei Shabbos.  But even so this minhag needs clarification.

The Leket Yosher brings from the Trumas Hadeshen that on Shabbos we have simcha from the Torah we learn.  Since the Shechina only rests when a person is happy and not sad, we start Slichos while the joy of Shabbos is still with us.  

Similarly the Vayaged Yaakov brings from Rav Yisachar Dov of Belz that it is hard for us to ask for Hashem to forgive us since our heads should be buried in shame.  However the Medrash says that Hashem's bracha to Shabbos was "Me'or Panim", a radiating face.  Therefore we hurry to start Slichos as close to Shabbos as possible so that we can show our face and not be embarrassed.  (Minhag Yisroel Torah 581:8)