The Venezuela midfielder bagged the game's only goal in the 58th minute to earn Lucien Favre's side their third win in five league games this term.
The result kept Gladbach two points behind new leaders Bayern Munich, while Kaiserslautern remained in the relegation zone having collected only two points so far.
The home team started brightly at Borussia-Park, Marco Reus and Igor de Camargo combining well before the latter's powerful shot was blocked by defender Martin Amedick.
It was Gladbach goalkeeper Marc-Andre ter Stegen who needed to make the first save of the match, though, to keep out a Kostas Fortounis effort in the 20th minute.
Thorben Marx went close to putting the hosts ahead 14 minutes later, his shot rolling just wide of Kevin Trapp's goal.
The deadlock was broken shortly before the hour when Tony Jantschke's shot flew through the penalty area and landed at the feet of Arango, who was unmarked at the far post and lashed home a volley which gave Trapp no chance.
Christian Tiffert almost pulled off a spectacular equaliser five minutes later with an overhead-kick, but Ter Stegen produced an equally brilliant save to divert the ball over the bar.
That was as good as it got for the visitors.
It could have got worse when Rodnei's poor backpass left Raul Bobadilla one on one with Trapp. The Argentinian's shot with the outside of his foot drifted narrowly wide, but Gladbach had already done enough.