After a goalless first leg it was expected to be a close game but it turned into a stroll for the five-times European champions thanks to the early dismissal of Oleksandr Kucher.
Bayern have now won all four home games in Europe this season, scoring 13 goals and conceding none as they set their sights on a sixth European crown.
“I always had faith with my players and I am very happy with their game tonight,” Bayern coach Pep Guardiola, whose team equalled their 7-0 win over Basel in 2012, told reporters.
“People said it may be a tricky game but my players are there for the big moments.
“Obviously the red card changed things but my players perfectly executed our game plan. Sometimes it is not that easy against 10.”
Bayern needed only four minutes to open their account after Mario Goetze was brought down by Kucher and Thomas Mueller converted the spot-kick.
It was the fastest red card in Champions League history from the start of a game and Mueller’s fourth goal this season, all from penalties.
The setback forced a complete change of plan for Shakhtar coach Mircea Lucescu but Bayern were relentless with Arjen Robben missing a huge chance after narrowly failing to connect with a Robert Lewandowski cutback in front of an empty goal.
The Dutchman’s thundering drive then sailed over the bar before he went off with a thigh injury after 19 minutes with a thigh injury.
The substitution did nothing to take the pace out of the Germans’ attacks, with the hosts hitting the post from a Robert Lewandowski header.
Jerome Boateng did better in the 34th, tapping in on the rebound after a Lewandowski shot was temporarily cleared with Shakhtar’s defence in complete disarray.
The Bavarians picked up after the break where they left off with two quick goals from Franck Ribery, who also went off injured, and Mueller.
The Germany international joined Mario Gomez on 26 goals as the top all-time German scorers in the competition.
Badstuber scored his first goal since coming back from injuries that ruled him out for two years before Poland striker Lewandowski also got on to the scoresheet.
Goetze made sure of a mention in the record books with the seventh goal of the evening.
(Reporting by Karolos Grohmann, editing by Ed Osmond)