The Wolves, who held a 1-0 lead from the first leg, finished second to champions Bayern Munich just two years ago, but they found themselves in the relegation playoffs after ending the regular season third from bottom.
Daniel Didavi came close to opening the scoring in the early stages of the clash when his low strike was aimed towards the net, but goalkeeper Jasmin Fejzic blocked the shot.
Mario Gomez, who scored the only goal in the first leg, thought he had found the breakthrough this evening when he scrambled the ball into the back of the net in the 39th minute, but it was correctly ruled out for a push in the box.
Just before the half-time break, Ken Reichel came close to nudging the Lions ahead, but his attempt from inside the 18-yard area flew over the crossbar.
The deadlock was eventually broken in the 49th minute when Vieirinha unleashed a first-time shot from the edge of the box and the ball nestled into the top corner.
Things went from bad to worse for Braunschweig when Maximilian Sauer was given his marching orders after receiving a second yellow card.
Wolfsburg had an opportunity to double their lead when Luiz Gustavo got inside the six-yard box, but he skied the ball over the box, and this came just minutes after Philipp Wollscheid's effort was ruled out for offside.
Braunschweig last competed in the Bundesliga in the 2013-14 season, but were relegated after just one campaign back in the top flight.