The Kansas City Chiefs in Super Bowl I set a the initial low mark with 10 points against the Green Bay Packers. This mark only stood for two years as Weeb Ewbank's defense held the heavy favorite Baltimore Colts offense to just 7 points in the biggest upset in Super Bowl history. The next year, in Super Bowl IV, the Minnesota Vikings equaled the Colts previous low by putting up just 7 points against the Kansas City Chiefs. Two years later the Miami Dolphins set the current Super Bowl's poorest offense output by putting up just 3 points against the Dallas Cowboys. The Dolphins did redeem themselves the next year by going undefeated, so I'm pretty sure this negative record doesn't bother them much.
To my semi-surprise there has yet to be a shut out in the Super Bowl.