PSG are on 69 points with four games left with Monaco trailing by 11 points
City had been largely frustrated throughout the 90 minutes and the extra period as Chelsea grew into the game.
But it was Pep Guardiola's City who proved more clinical with their penalties and Raheem Sterling struck the decisive blow past Chelsea keeper Kepa Arrizabalaga.
Extra time had finished in farcical circumstances when Spaniard Arrizabalaga refused to be substituted after appearing to be injured making a late save from Sergio Aguero.
City have now won the League Cup four times in six seasons to move to second on the all-time list with six titles, two behind the record eight of Liverpool.