The hosts took an early two goal lead when Dominick Kumbela tapped in unmarked in the 14th minute and Havard Nielsen broke free seven minutes later to double their advantage.
An 89th minute goal by Jan Hochscheidt sealed the win in the Lower Saxony derby, which had a strong police presence on alert for fear of crowd trouble.
The win lifted 18th-placed Braunschweig to 25 points, two off 15th place and safety and landed Hanover, who had Andre Hoffmann sent off on the hour for kicking an opponent, in the relegation mix in 13th with 29 points.
Nuremberg are in 17th place on 26 points with Hamburg SV and VfB Stuttgart on 27.
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In the other match on Sunday Hertha Berlin drew 1-1 against Hoffenheim with the result denting both clubs' chances of playing in Europe next season.
Hertha's Sami Allagui pounced on a defensive error to curl in a low drive in the 11th minute but Eugen Polanski drew the visitors level, completing a swift passing move on the half hour
Hoffenheim, who are likely to be without Belgian keeper Koen Casteels for some time after he was taken off by stretcher with a suspected fractured leg, are ninth on 37 with Hertha a spot behind on goal difference.
Champions Bayern Munich saw their 53-game unbeaten run in the league snapped when they lost 1-0 to Augsburg on Saturday.
Borussia Dortmund tightened their hold on second place with a 2-1 win over VfL Wolfsburg that kept them three points clear of Schalke 04.
(Reporting by Karolos Grohmann; editing by Josh Reich)
