New signing Henrikh Mkhitaryan made his top-flight debut for Dortmund, with last season's runners-up frustrated for much of the game as they struggled to break down the visitors' defence.
However, Jurgen Klopp's men eventually edged ahead thanks to second-half goals from two substitutes.
Jonas Hofmann, who came on for Pierre-Emerick Aubameyang, was the first to strike after 75 minutes before Jakub Blaszczykowski's replacement, Marco Reus, added another from the penalty spot.
Kevin Kratz grabbed a late goal for the visitors from a corner but this proved to be no more than a consolation.
The result leaves Dortmund top of the pile, while Braunschweig are still waiting for their first win back since their return to the Bundesliga.
Sunday's other game saw a four-goal thriller at the Grundig-Stadion, as Nuremberg and Hertha Berlin drew 2-2.
After their 6-1 win against Eintracht Frankfurt a week ago, Hertha found themselves a goal down at half-time thanks to a Josip Drmic effort in the 40th minute.
However, it looked as though the newly promoted side may pick up consecutive wins when Sami Allagui and Ronny put them 2-1 ahead.
It was not to be though, as Hiroshi Kiyotake struck a minute from time to claim a second consecutive draw for his side.
Nuremberg now sit in 11th place after two games, while Jos Luhukay's side are sixth.
