It was the visitors who made the most impressive start with their pack firing on all cylinders and they opened the scoring with a Calvin Howarth penalty.
However Edinburgh battled back into the contest in the second quarter, firstly Brendan Laney levelling the score, then their defence holding up as both Kellock and Lee were sent to the sin bin.
Howarth struck a second penalty early in the second half but within three minutes Southwell had squeezed in at the corner for Edinburgh and the game's first try.
Howarth completed his penalty treble but Southwell retook the initiative sending a beautifully weighted kick towards the line and as three defenders struggled to make it safe Craig Joiner nipped in to touch down, with substitute Phil Godman adding the awkward conversion.
Even better was to come for the Gunners with Godman banging over a close-range penalty which also resulted in Donnie Macfadyen being sent to the sin-bin.
Dan Parks slotted a late penalty to salvage a consolation bonus point for Glasgow.
Ali Hogg celebrates Edinburgh's win with Hugo Southwell