The Bosnia and Herzegovina international scored three times in a devastating 12-minute spell to give Stuttgart a commanding lead before half-time and put the German side firmly in control of their first Champions League match of the season.
• VfB Stuttgart: Ermedin Demirović 20', 26', 32'
• Viking Stavanger: Zlatko Tripić 22'
Stuttgart made the stronger start and were rewarded after 20 minutes when Demirović finished from close range following good work from Deniz Undav.
Viking responded almost immediately. Zlatko Tripić punished a defensive lapse two minutes later to restore parity and briefly give the visitors hope of causing an upset.
However, Stuttgart quickly regained control.
Demirović restored the hosts' advantage in the 26th minute before completing his hat-trick six minutes later with another clinical finish. His three goals came inside just 12 minutes, turning an evenly poised contest into a comfortable Stuttgart lead.
The German side went into the interval 3–1 ahead and continued to dominate after the restart.
Viking struggled to create a sustained response, while Stuttgart had opportunities to extend their advantage. Deniz Undav, who played a key role in the attacking build-up and provided two assists, was among those to threaten the visitors' goal.
Stuttgart ultimately settled for the 3–1 victory, giving Sebastian Hoeneß's side an ideal start to their Champions League campaign.
The result also marked Stuttgart's return to Europe's premier club competition after a 588-day absence. Their previous Champions League appearance had ended in a 4–1 defeat to eventual champions Paris Saint-Germain.
For Demirović, however, the night belonged entirely to him. His hat-trick immediately put him among the leading scorers in the new Champions League league phase and gave Stuttgart a major early boost in their attempt to reach the knockout stages this time around.
Stuttgart will next turn their attention to Bundesliga action before returning to Champions League duty against Slovan Bratislava in October.