The Armenian, who completed his switch to the Bundesliga side on Monday, set up Marco Reus to score an early goal before producing a fine left-footed finish after Fabian Schar had levelled for the hosts on Wednesday.
And, with ex-Shakhtar Donetsk playmaker Mkhitaryan long since withdrawn as one of numerous substitutions, Jonas Hofmann completed the scoring for Juergen Klopp's men with some fine control and a cool finish in the second period.
Dortmund named a strong team ahead of the fixture at St Jakob Park, with recent arrivals Sokratis Papastathopoulos and Mkhitaryan backed by familiar faces such as Roman Weidenfeller, Nuri Sahin and Robert Lewandowski.
A combination of old and new soon brought about the first goal, with Mkhitaryan bursting through on the right before teeing up Reus for an easy tap-in from close range.
But Basel were soon back on terms, as a free kick from the right was nodded in by a grateful Schar.
There was still time for Mkhitaryan to put a further stamp on the half, however, as he linked up brilliantly with Sahin before half-volleying into the far corner from the left.
Klopp made nine changes to his team at the break and the game lost some of its fluency as a result.
However, the 2013 Champions League finalists rounded off their victory with a moment of brilliance in the dying moments.
Hofmann pulled off a fantastic touch to push a cross-field ball past a despairing Basel defender before showing the presence of mind to slot the ball home and complete the win.