Ken Pisi thought that he had given Northampton the lead midway through the opening half, however the video referee was unconvinced that the ball had been grounded legally.
Sean Lamont then scored against his former club to give Glasgow a 15-0 lead with 12 minutes of the first half remaining.
Pisi doubled his tally in the dying seconds of the opening period, collecting a clever pass from his brother before grounding between the posts.
The turnaround was completed by a Stephen Myler try three minutes after the restart.
Vasily Artemyev then earned Northampton a bonus point when he crossed the line 10 minutes from time.