Citizens head coach Pep Guardiola made four changes to the side beaten 1-0 at Liverpool, with both David Silva and Sergio Aguero left on the bench.
The Clarets welcomed midfielder Jeff Hendrick back after suspension, with Steve Defour dropping to the bench, with Johan Berg Gudmundsson also given a start.
The hosts enjoyed the better of the early possession and seven minutes in visiting keeper Tom Heaton did superbly to deny Kelechi Iheanacho and then Raheem Sterling from the follow-up.
Iheanacho saw an effort deflected behind for a corner as City started to turn the screw, with Heaton then forced to turn away a Yaya Toure effort from the edge of the area.
Striker Iheanacho was crowded out inside the Clarets area just before the half-hour mark, but the hosts were reduced to ten men soon after as Fernandinho was shown a straight red card for a two-footed lunge on Gudmundsson.
Guardiola threw on both Aguero and Silva at half-time to try and give his side some inspiration and City began the second 45 minutes brightly.
Sterling saw an effort blocked and the loose ball almost fell to Aguero as the hosts continued to press, with the visitors still looking dangerous on the counter.
However, the Clarets were undone just before the hour mark as Clichy cut inside on the left side of the area and his low, right-foot shot found the far corner of the net.
It was soon 2-0 as Sterling fell in trying to round Heaton and Aguero pounced to smash home a shot from a narrow angle, beating two defenders who were covering on the line.
Burnley pulled a goal back 20 minutes from time when Ben Mee rattled home a close-range effort following a corner from the left.
The Clarets then pushed hard in the closing minutes to try and find an equaliser, with Michael Keane seeing an effort tipped over in stoppage time by keeper Claudio Bravo.