Augsburg appeared to have no answer to a bright start from the visitors, who doubled their lead before the half-hour mark as Peter Niemeyer reacted quickest in the box to score after Marwin Hitz had denied Luca Caldirola.
After struggling to halt Darmstadt's dominance in the opening 45 minutes, Augsburg came close to reducing the deficit after the break when Raul Bobadilla fired just wide from the edge of the box.
Bobadilla looked to be the likeliest candidate to get the home side back in the game, but he wasted a great chance by failing to find the target from close range after being picked out by Daniel Baier's cross.
Darmstadt never appeared to be too concerned by a possible fightback, and they continued to threaten as the opposite end as the influential Rausch curled wide of the post with a long-range effort.
It proved to be a comfortable finish for the visitors as they protected their unbeaten away record to pile more misery on Augsburg's supporters.