Sean Maitland went over for his first Warriors try early on with Edinburgh hitting back soon after through Greig Laidlaw's penalty.
Laidlaw missed his next penalty before Ruaridh Jackson intercepted and burst forward to cross with two Pete Horne penalties making it 16-3 at the break.
Soon after the break DTH van der Merwe saw his potential try ruled out by the TMO with his foot in touch but Pyrgos was ruled to have touched down in the 55th minute.
Tries from Roddy Grant and Tim Visser, with Laidlaw converting both, got Edinburgh back in it but the Warriors held on for a 44-31 aggregate victory.
Glasgow opened the scoring in the seventh minute with some composed work by Jackson who picked out Maitland to touch down.
Pete Horne slid his conversion wide before Edinburgh hit back in the ninth minute through a Laidlaw penalty.
Horne then found himself in the sin-bin just before the quarter-hour mark for a tackle off the ball and Edinburgh failed to capitalise with Laidlaw missing a 22nd-minute penalty.
And just before the half-hour mark the Warriors made Edinburgh pay with Jackson intercepting Piers Francis' pass to burst clear and run through for a try but Horne's conversion from a tight angle went wide of the post.
Three minutes before the break Horne made no mistake with his penalty to put the Warriors 13-3 up and when Edinburgh were penalised at the breakdown on the stroke of half-time Horne kept his nerve again.
Soon after the interval, Van der Merwe appeared to go over in the corner after good work by Maitland but the TMO ruled he had a foot in touch.
The TMO was called on again in the 55th minute when the ball came out of the scrum and Pyrgos pounced on it to touch down but this time it was ruled good.
Horne missed his conversion again and just before the hour Laidlaw dinked the ball over the top with Matt Scott carrying and offloading to Grant to score, with Laidlaw converting to make it 21-10.
Minutes later Stuart Hogg spilled what looked to be a certain score for the Warriors over the line.
Visser then crossed to make himself the joint top scorer in the RaboDirect PRO12 of all time with Laidlaw converting with five minutes remaining to make the scores 21-17 and ensure a tight finish, but the Warriors saw it out.
|Edinburgh Rugby Score Card|
|Glasgow Warriors Score Card|