Lionel Messi came on as a substitute for the last 30 minutes for his first appearance of the season but was unable to spark the visitors' struggling attack, with Dortmund creating the best chances of the game.
Ter Stegen saved a penalty from Dortmund captain Marco Reus in the 57th and denied the home side with a string of saves on his return to his native Germany.
Dortmund substitute Julian Brandt struck the crossbar and Paco Alcacer also went close against his former side. It was the first game in which the Spanish striker failed to score this season.
The 16-year-old Ansu Fati made his Champions League debut to become Barcelona's youngest player in a Uefa competition, before coming off for Messi in the 59th. Messi hadn't played since injuring his leg in pre-season.
Defender Mats Hummels showed why Dortmund brought him back from Bayern Munich with several vital interventions, starting with a block to deny Antoine Griezmann's first effort for Barcelona, then a crucial headed clearance from the lurking Luis Suarez. Griezmann, on his Champions League debut for Barcelona, produced the cross.
Dortmund's first big chance fell to Reus, who was denied for the first time by ter Stegen in the 25th. England international Jadon Sancho also missed a good opportunity.
A Barcelona corner caused chaos in the Dortmund defence, with Roman Burki punching the ball clear and Fati's effort from the rebound deflected over by Reus.
Dortmund were more aggressive after the break but ultimately couldn't get past ter Stegen. Messi, meanwhile, looked far from his best and the home fans cheered loudly when he was crowded off the ball by three defenders as the Argentine failed to make an impact.
The 19-year-old Norwegian gave Salzburg the lead in the second minute and added two more in the 34th and 45th. Haaland is the third-youngest player to score a Champions League hat trick after Raul in 1995 and Wayne Rooney in 2004.
The son of former Leeds and Manchester City player Alf-Inge Haaland, he attracted attention in May when he scored nine goals for Norway in an under-20 World Cup game against Honduras. Haaland has torn up Austrian competitions since joining in the summer from Norway's Molde, scoring 14 goals in eight league and cup games.
Hwang Hee-chan scored one goal and set up two more for Salzburg, who are playing in Europe's top club competition for the first time in 25 years. It was the highest-scoring Champions League game since Paris Saint-Germain beat Celtic 7-1 in 2017.
The Austrian champions have not lost any of their last 19 European home games in deep runs through the Europa League knockout stages and has won nine of its last 10.
Belgian champions Genk have played in the Champions League group stage twice before, in 2002 and 2011, but have failed to win any of their 13 games to date.
Champions League results
- Red Bull Salzburg 6-2 Genk 2
- Napoli 2-0 Liverpool
- Inter Milan 1-1 Slavia Prague
- Borussia Dortmund 0-0 Barcelona
- Lyon 1-1 Zenit St Petersburg
- Benfica 1-2 RB Leipzig
- Ajax 3-0 Lille
- Chelsea 0-1 Valencia
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