The disciplinary committee said his punishment would have been longer had he not shown so much remorse.
It explained an offence of playing an opponent without the ball should have resulted in an "entry-point of a five-week playing suspension".
But the committee granted Hogg a "40% reduction" in his punishment to "account for mitigating factors, including the player's remorse for the incident both on the day of the match and subsequently through the media".
That means that the 21-year-old will miss Glasgow's clashes with Scarlets, Ospreys and Treviso and the full-back repeated his regret over his actions.
"I'm very sorry," he said. "I have let people down and apologise to my team-mates, the Scotland management and all Scotland supporters.
"I am disappointed to be missing three games but fully respect the committee's decision."