Gladbach, who have lost all three away games this season but have scored 11 goals in their three home wins, took the lead with Oscar Wendt's powerful drive in the 22nd minute before doubling the score through Brazilian Raffael nine minutes later.
But their hopes were dashed when Gladbach struck twice in three minutes. Juan Arango was brought down in the box in the 72nd minute and Germany international Max Kruse converted the spot kick before Raffael snatched his second goal of the evening and his third this season in the 75th.
Gladbach move up to fourth on nine points with leaders Borussia Dortmund, on 15, in action against Nuremberg on Saturday.
Braunschweig, back in the Bundesliga after a 28-year absence, are in last place on one point.
Champions Bayern Munich in second place on 13, travel to fellow Champions League competitors Schalke 04 on Saturday.