The Citizens were a goal to the good and cruising after a dominant start to the second period, but they were pegged back inside the final 10 minutes through a Johann Berg Gudmundsson goal.
Gudmundsson's equaliser, which prevented City from going a whopping 18 points clear at the top, came shortly after a contender for miss of the season from Raheem Sterling, as he failed to convert into an empty net from inside the six-yard box.
Without a number of key personnel, City took some time to get going in Lancashire with a few speculative shots amounting to little prior to their breakthrough a quarter of the way through.
The league leaders, boasting the joint-best record of any English top-flight side after 25 games, started to click into gear but could not add a second to their tally before the break.
Sergio Aguero wasted his best opening at the end of the opening 45 minutes by shooting at Pope, and it was the Clarets - winless in 51 games when falling behind prior to today - who were left to rue not going into the break all square.
A couple of chances came and went for Ben Mee, with a left-foot volley pushed aside by Ederson and a back-post header flying wide of the target, though Pope also had to be alert to block Kevin De Bruyne's long-range shot.
Aiming for a first away league win in three, City almost added a second goal early in the second half as Aguero blasted into the back of substitute Matt Lowton from a good position and Ilkay Gundogan's attempt deflected over.
City were perhaps guilty over overplaying at times as Aguero passed up the chance to shoot on a couple of occasions, though they were creating plenty of openings and should have wrapped things up.
Danilo's latest curler on the hour mark, again following a short corner, was this time a little more central and tipped over the bar by Pope.
Having sat exclusively in their own half for the opening 20 minutes of the second half, the hosts suddenly got at their opponents and would have been level if not for a moment of Ederson brilliance.
Moments later Sterling produced one of the great Premier League misses, tapping the ball wide right in front of a half-empty goal after being picked out by Kyle Walker.
That miss would prove costly as Gudmundsson met a cross at the back post and sent his half-volley past Ederson, meaning a third successive away game in which City have dropped points.
BURNLEY (4-2-3-1): Pope; Bardsley (Lowton 46'), Long, Mee, Taylor; Cork, Hendrick; Lennon, Barnes, Gudmundsson; Vokes
MANCHESTER CITY (4-3-3): Ederson; Walker, Kompany, Otamendi, Danilo; Fernandinho, Gundogan, De Bruyne; Bernardo, Aguero, Sterling (Diaz 74')