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.