PSG goalkeeper Keylor Navas played a key role with a string of saves as Bayern peppered his goal late on in a desperate bid to grab an equaliser.
This was a record 19th appearance in the Champions League quarter-finals for Bayern, but there was little for the Bavarian giants to celebrate at the final whistle.
Covid-19 cast its shadow over both teams with Bayern winger Serge Gnabry, plus PSG pair Alessandro Florenzi and Marco Verratti quarantined after testing positive.
Heavy snow fell in the first half in Munich as the visitors put the heat on the holders.
The opening goal came after Neymar sprinted through a gaping hole in the defence and laid the ball off for Mbappe, whose shot flew into the net through Manuel Neuer’s legs.
PSG’s Germany forward Julian Draxler had a goal disallowed with Mbappe offside in the build-up as the visitors kept up the pressure.
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