The French League champions went 3-1 up thanks to a brace from former Man United striker Falcao before City fought back.
Two goals from Sergio Aguero and one each for Leroy Sané, John Stones and Raheem Sterling gave City a 5-3 lead to take to the second-leg in Monaco.
Manager Pep Guardiola says it's an important win
Meanwhile Atlético returned to Madrid with a 4-2 advantage after their first leg against Bayer Leverkusen in Germany.
