Gregor Townsend's men started slowly but got their act together to consolidate second place in the table.
Warriors were rattled when they quickly went seven points down.
But they replied with a fine solo try from Byron McGuigan, which Scott Wight converted.
The home side were back in front when a Wight kick was charged down and Teo Tebaldi went over for a converted try.
But Glasgow wasted no time in responding, Alex Dunbar powering over the try line, Wight gaining the extra points.
Then he banged over a penalty to give the visitors the lead.
Zebra squared the game with a penalty. And they repeated the dose a little later.
Both teams were reduced to 14 players when Warriors Graeme Morrison and Zerbe's Filippo Ferrarrini were yellow carded.
That was the cue for the Warriors to step up the pace and they were rewarded when McGuigan completed his double following a pass by Ryan Wilson.
Then Glasgow's James Eddie claimed the crucial fourth touchdown.
Rookie substitute Finn Russell converted to stretch the gap to nine points.
And Pete Murchie snapped up try No.5 finishing off a great move featuring Niko Matawalu and Tim Swinson.
Russell again added the extras.