Sean Lineen's troops trailed 16-11 at the break after two tries from Scottish winger Nikki Walker and two Dan Biggar penalties, with Richie Gray crossing over for the hosts.
But the Warriors staged a rousing fight back to secure the points and move up to tenth in the standings thanks to scores from Richie Vernon and Kevin Tkachuk, despite Walker completing his hat-trick late-on.
Blindside flanker Barclay, 24, was delighted at the manner in which his teammates kept their cool despite a poor first half, and insisted there was no reason their heroics could not be repeated this Friday against the Scarlets.
"That's one of the best second-half performances we've produced in the seven years I've been with Glasgow," said Barclay, who has 20 Scotland caps to his name.
"We had a couple of defensive lapses early on but we didn't panic. I felt we were playing good stuff even at that stage and we were getting quick ball against a team that's hard to get quick ball against.
"We've got to take a lot from that while keeping our feet on the ground because there are not many superstars in our team but we've won at the Scarlets a couple of times in the last few years and why not if we play like that?"