Tuchel replaced Jurgen Klopp in charge of the Bundesliga side in July.
Armenia midfielder Henrikh Mkhitaryan scored twice for Dortmund while Marco Reus and Pierre-Emerick Aubameyang were also on target.
Dortumund's win came after champions Bayern Munich had cruised to a 5-0 victory over Hamburg on Friday evening.
Elsewhere on Saturday, Bayer Leverkusen warmed up for their Champions League play-off clash at Lazio on Tuesday with a 2-1 win against Hoffenheim.
Newly promoted Ingolstadt celebrated their debut in Germany's top flight with a 1-0 win at Mainz .
Darmstadt's first Bundesliga match for 33 years ended in a 2-2 draw against Hannover in an action-packed match.
Schalke picked up their first win under new coach Andre Breitenreiter with a 3-0 victory at Werder Bremen, while Hertha Berlin won 1-0 away to Augsburg.
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