Pep Guardiola's men began the 2021/22 UEFA Champions League group stage in perfect fashion, with Nathan Ake powerfully heading in from Grealish's curling corner just after 15 minutes, marking City's 100th goal in the competition.
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Shortly before the half-hour mark, City were celebrating a second, as Kevin de Bruyne's sensational cross was turned into his own net by Nordi Mukiele.
An entertaining encounter at the Etihad took a twist just before the interval as Leipzig pulled a goal back through Christopher Nkunku - who headed beyond Ederson from six yards.
With the clock ticking into stoppage time at the end of the first half, City were awarded a penalty after a handball by Lukas Klostermann and Riyad Mahrez emphatically fired into the top corner to restore the two-goal lead.
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The second period began with a similar frantic pace and the visitors reduced the deficit for a second time, with Nkunku firmly heading in his second of the evening five minutes after the interval.
City struck back almost immediately again, with Champions League debutant Grealish driving inside from the left flank and curling a superb shot into the corner to make it 4-2.
Once again, the Bundesliga side dragged themselves back into the game, with Nkunku completing his hat-trick with a clever finish from a tight angle in the 73rd minute.
Two minutes later, City opened up a two-goal lead again, with Joao Cancelo arrowing a shot into the corner from 25 yards.
With just over ten minutes remaining, former City fullback Angelino was sent off for a second bookable offence after bringing down Cancelo. Five minutes later, substitute Gabriel Jesus scored City's sixth of the evening, striking into the roof of the net at the back post to confirm City's three points.
View the best images from the encounter via our gallery below.