Domi Kumbela, Havard Nilsson and Jan Hochscheidt were on target as Torsten Lieberknecht's men completed the double over their local rivals and moved to within two point of safety with their biggest win of the season.
Hannover, who ended the game with 10 men after Andre Hoffmann was shown a red card, now have to look over their shoulders heading into the final five games of the season, with only two points separating them from the bottom three.
Spice was added to the local derby in Lower Saxony by the fact that both clubs needed the points in their respective battles against relegation.
Braunschweig, urged on by a partisan crowd, went close to opening the scoring in the 12th minute when Kumbela sent his shot over from 12 yards out.
Two minutes later Kumbela was in the right place at the right time to tuck the ball in after Ron Robert Zieler pushed Ken Reichel's cross out to his feet.
Seven minutes later Braunschweig doubled their lead when a long ball from deep landed in the path of Nielsen and he let it bounce just once before sweeping it past Zieler with a fine left-footed volley into the far corner.
Hannover tried to respond, but their hopes were dealt a setback when Hoffmann kicked out at Mirko Boland, referee Peter Gagelmann seeing the offence and producing a red card.
Despite being a man down, Hannover did not throw in the towel and almost pulled a goal back when Szabolcs Huszti's shot came back off a post.
It was not to be, though, and, as their belief in a recovery waned, they conceded a third.
Karim Bellarabi squared the ball for Hochscheidt, who crowned the win with the final goal in the 89th minute.