Former Germany international Mike Hanke's glancing header in the 31st minute put them on track for their seventh win in the last nine league games with Marco Reus and Igor de Camargo on target as well late in the game.
Gladbach, who came through a relegation play-off last season to stay in the top flight, remain fourth and have 39 points from 19 games after their first win at Stuttgart since 1994.
"The result is a bit deceptive," Hanke told reporters. "We were not as dominant as in our home games. But we are satisfied and proud with this win.
"The title is absolutely no issue for us at the moment. We know where we come from, we are far from a top team. We have a stable team and we won 3-0 away today."
In Sunday's other game, Eric-Maxim Choupo-Moting scored twice in Mainz 05's 3-1 win over bottom-placed Freiburg that lifted them to 12th, three points above the drop zone.
