Šťastný began playing junior ice hockey in his hometown club HK 95 Považská Bystrica.

He also played junior hockey in MsHK Žilina and HK Dukla Trenčín.

He was selected to the HK Orange 20 project to preparation for the World Junior Championship in 2010.

In 2010 he moved to North America to play for the WHL club Vancouver Giants.

Before the 2011–12 season he came back to Slovakia, signing for HK Dukla Trenčín.