Stastny’s screened wrist shot beat goaltender Mike Smith high to the right corner.
Connor Hellebuyck made 38 saves for Winnipeg, and Smith stopped 35 shots.
Winnipeg won the opener 4-1 on Wednesday night. The series shifts to Winnipeg for games Sunday and Monday nights.
Oilers captain Connor McDavid and teammate Leon Draisaitl — the NHL’s top two scorers — have been held off the scoresheet in the first two games.