The former Liverpool midfielder fouled Taison to collect a second yellow card with around 25 minutes to play.
Earlier, Thomas Muller had pulled a shot wide when clean through for Bayern and he fired over before the break.
But neither side threatened after Alonso's red in a tetchy game. They meet again in Munich on 11 March.
The sides shared 35 fouls and six yellow cards in addition to Alonso's red, with attacks often cut short by tactical fouls when danger looked likely.
Shakhtar, champions in Ukraine, will be "heroes" if they overcome Bayern according to the manager Mircea Lucescu as his side are still playing home fixtures away from their Donbass Arena because of conflict in the area.
They have also not played a competitive game since December as the Ukraine Premier League - in which they sit second to Dynamo Kiev - is on its winter break.
Playing several hundred miles away from home in the western city of Lviv, they arrived to face a Bayern side who had hammered Hamburg 8-0 on Saturday but ended the night having restricted their visitors to few chances.
Though Muller sprang the offside trap from Arjen Robben's pass in the first half, he could only pull his shot across the goalmouth from 10 yards and he then fired over from a Franck Ribery pull-back late in the first period.
Shakhtar, boasting the tournament's top scorer in Luiz Adriano, played on the break but only got their talisman into the game sparingly and he left the field late on having completed just 44% of his 18 passes.
Alonso tangled with Shakhtar winger Taison when the Brazilian threatened a break away, prompting a second yellow card, but the Ukrainians failed to make their advantage count and both sides settled for a point.
They will reconvene in three weeks' time in Munich, where Bayern have not lost since defeat against Real Madrid in the semi-final of this competition last season.
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