Shakhtar coach Mircea Lucescu had been determined that his team would not make the same mistakes in the competition as they did last term, when the Ukrainian champions managed only one win in their group to finish bottom.
They never recovered from losing their opening game against Porto before being held 1-1 by perceived minnows APOEL Nicosia, but this time made no mistake against the Danish champions, the team widely viewed as the weakest in Group E.
Mkhitaryan opened the scoring just before half-time and grabbed his second 15 minutes from time to ensure his side top the group after Chelsea and Juventus drew 2-2 at Stamford Bridge.
Clear chances were hard to come by in the first half, with midfielder Willian having the first meaningful attempt after 17 minutes with a header which was saved by Jesper Hansen.
At the other end, Nicolai Stokholm's long-range drive was parried by Andrei Pyatov as the visitors tried to upset the odds.
But after Hansen had saved a 12-yard shot from Mkhitaryan on 39 minutes, the goalkeeper was finally beaten shortly before the interval as the same player fired into the bottom corner from 12 yards.
Hansen then pulled off a double save early in the second half to keep his side in the game, firstly blocking Darijo Srna's 20-yard volley and then reacting quickly to also keep out Tomas Hubschman's effort from closer range.
Ivan Runje then brought a save from Pyatov with a header from six yards at the other end, but Mkhitaryan was soon threatening again with a shot which Hansen parried away from danger.
Hansen also tipped over a 25-yard effort from Willian before his resistance was finally broken in the 76th minute as Mkhitaryan found enough space in the area to find the bottom corner again.
The visitors refused to throw in the towel but Hansen remained the busier goalkeeper in the closing stages, denying Marko Devic and Alex Teixeira as the home side looked to boost their goal difference.
