Sven Schipplock's five strikes in the first round of the DFB Cup last weekend were not enough to earn him a starting berth, but Roberto Firmino was back in the fold.
It was another of Markus Gisdol's attacking options who netted their first Bundesliga goal of the season, though, with Szalai scoring from close range as Augsburg struggled to clear their lines.
Sharp up front, solid at the back
The dust had barely settled when the hosts struck again. One of the sweeping attacks which they patented last season allowed Firmino to tee up Elyounoussi, who slid the ball past Marwin Hitz for his first Hoffenheim goal.
Augsburg, still reeling from defeat in the DFB Cup last weekend, tried to respond in the second half, but Hoffenheim would not be breached. They locked up at the back and showed they can also stay resolute in defence, which will come as a welcome boost to a side that shipped 70 goals last season.