Marcel Heller also found the net twice, including a spectacular 70-metre solo run for his first goal, as promoted Darmstadt 98 marked their return to the top flight after 33 years with a 2-2 draw at home to Hanover 96.
Dortmund, rock bottom midway through last season before a revival rescued a Europa League spot, showed they mean business under Tuchel as they gave Gladbach, third last season, no chance.
It proved a memorable day for Tuchel who is eager to shake off the shadow of predecessor Juergen Klopp who won two league titles with the club.
"Superb consistency and great goals. I am very satisfied and the team knows they did a good job," Tuchel told reporters.
"The foundations left behind by Juergen Klopp are outstanding. It was a pleasure seeing the team work so hard with such passion. It was an outstanding performance."
Marco Reus' unstoppable shot beat keeper Yann Sommer in the 15th minute and Pierre-Emerick Aubameyang doubled the lead six minutes later, heading in a Marcel Schmelzer cross.
With Dortmund keeping up their frenzied attacking pace it did not take long for them to score again, Mkhitaryan tapping the ball in at the far post.
OUT OF SORTS
The Armenian, who struggled last season, added another goal five minutes after the restart from a Reus cutback as the defence of Gladbach, who are through to the Champions League group stages, looked totally out of sorts.
Schalke 04 also began in style, crushing hosts Werder Bremen 3-0 to give new coach Andre Breitenreiter a winning start with the Ruhr valley club.
Werder's Theodor Gebre Selassie gave the visitors an unexpected lead when he chipped the ball over his own keeper and Eric-Maxim Choupo-Moting slotted in a superb Joel Matip through ball in the 68th minute.
Klaas-Jan Huntelaar completed the scoring after Leroy Sane charged forward and cut the ball back for the Dutchman who became the fourth highest Bundesliga scorer for the club with 69 goals in 129 appearances.
Bayer Leverkusen came from 1-0 down to beat Hoffenheim 2-1 with Stefan Kiessling and Julian Brandt on target after Steven Zuber had given the visitors a fifth-minute lead.
Promoted Ingolstadt stunned hosts Mainz 05 1-0 while Hertha Berlin edged past Augsburg by the same score.
Both teams finished with 10 men after the dismissals of Augsburg's Raul Bobadilla and Roy Beerens of Hertha.
On Friday, champions Bayern Munich demolished Hamburg SV 5-0.
(Editing by Pritha Sarkar)
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