The former Orlando Pirates man capped off his first start for the club with a 50th-minute strike at Olympia Stadium, having made his debut as a substitute last week.
Bafana Bafana coach Stuart Baxter has expressed hope that Erasmus can force his way back into international reckoning with regular game-time after a frustrating spell for the SuperSport United product.
Erasmus joined Eskilstuna, who were relegated from the Swedish topflight in 2017 and are currently fourth on the table after four matches, in March after being released by Ligue One side Stade Rennais.
The 27-year-old only played two games for Rennais' reserves in France's fourth tier in 2017-18.
Erasmus was loaned to Ligue 2 outfit Lens last term and made 12 appearances.