The Glasgow Warriors player was dismissed 23 minutes into the Millennium Stadium clash in Cardiff after smashing his shoulder into the jaw of Dragons fly-half Dan Biggar.
Hogg - whose team went on to suffer their heaviest ever defeat in the tournament after being reduced to 14 men as Wales stormed to a huge 51-3 victory - will not be freed to play again as of midnight on April 6.
Hogg learned his fate after appearing before a disciplinary hearing in London and will now miss his club's next three RaboDirect Pro12 fixtures against Llanelli Scarlet, Ospreys and Benetton Treviso.
The 21-year-old apologised to Biggar, his team-mates and referee Jerome Garces - who initially showed him a yellow card before upgrading the punishment after the incident was replayed on the stadium's giant screens - after the match.
He is only the third Scotland player to be sent off in a Test match, following Nathan Hines in 2002 and Scott Murray in 2006.
In a statement, the Six Nations disciplinary committee explained Hogg's punishment would have been longer had he not shown so much remorse.
It said: "Stuart Hogg, Scotland's full back, appeared today in London before an independent Six Nations Disciplinary Committee, having been dismissed with a red card during the RBS 6 Nations match between Wales and Scotland on Saturday.
"The player did not contest the red card. The Disciplinary Committee, chaired by Mark McParland (Ireland), along with Jeremy Summers (England) and Achille Reali (Italy), having viewed the TV footage of the incident and considered representations by and on behalf of the player, determined that the player's actions constituted an act of foul play contrary to Law 10.4(f) [which relates to playing an opponent without the ball] and that it was in the mid-range of the IRB's sanctions for the type of offence, resulting an entry-point of a five-week playing suspension.
"The Disciplinary Committee found that there were no aggravating factors and granted the player a 40 per cent (two week) reduction 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.
"The Disciplinary Committee therefore imposed a sanction of a three week playing suspension. Stuart Hogg is suspended from playing until midnight on 6 April 2014 and has the right of appeal."