This "common wisdom" doesn't hold water.
With no changes to my eating habits, I started cycling each day on June 25th, About 12 hours per week of cycling on average (~5000 Calories per week).
I've been tracking my weight and body fat on a Health Mate scale. As of July 16th (21 days later), my:
• weight decreased by 3.4 lbs
• body fat went from 22 lbs to 17 lbs
A better statement of the "common wisdom":
You can lose weight by exercise alone, but only if you are willing to exercise enough to make a difference.