PSG missed several chances and were thwarted by Real keeper Thibaut Courtois, who denied Mbappe on several occasions before superbly pushing away Messi's spot-kick in the second half.
Real Madrid finished the game with zero shots on target while PSG had eight, with almost 60% of possession.
Real will be without Casemiro and Ferland Mendy for the second leg at home on March 9. The pair were both shown yellow cards and will be suspended for collecting three yellows in the competition.
Courtois repeatedly frustrated Messi and Mbappe. His best save came early in the second half when Messi stepped up for the penalty after Mbappe had been fouled inside the area by Dani Carvajal. The Belgian stretched low to his left to stop the Argentinian's shot with his hand.
But 30 minutes later Mbappe found the winner and Courtois could only watch as the ball went through his legs and into the corner of the net.