Ibisevic had a glorious opportunity to give the visitors the lead inside two minutes, but the striker fired straight at Marjan Petkovic after good work by Traore.
Braunschweig came close twice in the first half as they slowly clawed their way back into the game, with Domi Kumbela firing wide and Marco Caligiuri being thwarted by Thorsten Kirschbaum.
For all the home side's pressure, Ibisevic gave Stuttgart the lead against the run of play in the 40th minute with a deft touch on a teasing cross.
One soon became two when Maxim tapped home in the 50th minute after Petkovic could only palm Traore's powerful effort into the path of the onrushing midfielder.
Traore himself got on the scoresheet with a delightful, curled finish after playing a one-two with Christian Gentner, and the Guinean still had time to tee up Harnik for his side's fourth goal in the 88th minute.
The result leaves Braunschweig stranded at the foot of the Bundesliga table and without a win in their opening seven games.