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The 19-year-old showed why Dortmund paid Salzburg 20 million euros ($22 million) for him last month when he came on in the 56th minute with his new team 3-1 down.
"I am pretty relaxed about it... I don't really know why," said Haaland.
"I'm at a fantastic club with great team-mates and good people around me. I came here to score goals and it was a good debut for me."
Dortmund's victory left them fourth in the Bundesliga table, level on points with defending champions Bayern Munich and Schalke, four points behind leaders RB Leipzig, who host Union Berlin later.
Having already scored 28 goals in 22 games for Salzburg this season, Haaland needed just 183 seconds to open his Dortmund account, then added two more in quick succession for his sixth hat-trick of the campaign.
"He is still young and we would advise against expecting too much, but he brings us a lot of strength and has an incredible mentality," said Dortmund sporting director Michael Zorc.
Following Haaland's stunning entrance, Dortmund scored two goals in as many minutes as he fired in off the post, before England winger Jadon Sancho netted from a tight angle to equalise.
Haaland put Dortmund ahead for the first time with 18 minutes left when he tapped home after Thorgan Hazard got in behind the Augsburg defence, with the goal awarded despite a VAR review for offside.
Haaland completed his treble when captain Marco Reus played him into the area, slamming the ball home to make it 5-3.
It had been a vastly different game until Haaland's stunning entrance.
Augsburg deserved to be in control, taking advantage of poor Dortmund defending with the opening goal from striker Florian Niederlechner before Marco Richter's stunning long-range strike 20 seconds into the second half.
The shock of falling two goals down woke up the Dortmund attack as Germany winger Julian Brandt fired home their first goal off the crossbar just four minutes into the second half.
The game became an end-to-end affair and Niederlechner grabbed Augsburg's third by tapping in.
But it was one-way traffic thereafter as Dortmund coach Lucien Favre brought on Haaland for defender Lukasz Piszczek.
"He moves well between the lines, gives us more attacking possibilities and at 19 years old, he has a huge potential for improvement, that's the most important thing," said Favre.
However, he warned Haaland is not fit enough yet to play a full match.
"He hardly trained over Christmas with a knee injury, so he is not yet ready to play 90 minutes."
Elsewhere, South Korean Kwon Chang-Hoon and Nils Petersen scored Freiburg's goals in a 2-1 win at Mainz which moved their side sixth.
Werder Bremen kept a clean sheet for the first time this season as an own goal by Florian Kastenmeier secured a 1-0 win at Fortuna Duesseldorf.
Colombian striker Jhon Cordoba scored twice as Cologne pulled further away from the bottom three with a 3-1 win at home to Wolfsburg.
US international Timothy Chandler scored the winner as mid-table Eintracht Frankfurt won 2-1 at Hoffenheim.
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