The Clarets were struggling to create openings but when Jack Cork looped the ball into the area, it fell to defender Ben Mee whose volley forced a good save from Ederson.
Physio Ally Beattie was over-worked as he then had to revive goalkeeper Nick Pope after a collision with City captain Vincent Kompany as the visitors probed for the opening goal.
There is a fair chance that people will meet in pubs years from now to discuss how Sterling contrived to knock the ball wide of goal from three yards out.
However, it was Sterling's failure that will grab all the attention and he's surely nailed down the miss of the season prize for his effort.
But there are aspects about Guardiola's side that grate on the Portuguese, such as what he feels is a proclivity to go down. We need to score the second, third and fourth when we have the chance.
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Fellaini lasted only seven minutes as a second-half substitute before having to be withdrawn as United were beaten 2-0 at Tottenham in the Premier League on Wednesday. "We just missed the last moment to try to score the second one and playing away, in the last 10 minutes, anything can happen".
Burnley's back four, despite the absence of James Tarkowski, was well drilled and it required patience to break them down.
"We hope that soon they start protecting our players because the way other teams are tackling our players is not part of the game, it's not fair play and they have to try to punish the players that play this way". Pep Guardiola is in such an irascible mood today, as his Man City team travel to play Burnley in the Premier League.
"I was really pleased". This is why we dropped two points. That changed the feeling, the stadium erupted, everyone came with us. "I have a little idea about the players who are going to join us in the States but we are going (to) see". This is the club's first season at this level for a long time and we are discovering how tough it can be. But we still have some fine quality and we show that with goal.
Aiming for a first away league win in three, City nearly added a second goal early in the second half as Sergio Aguero blasted into the back of substitute Matt Lowton from a good position and Ilkay Gundogan's attempt deflected over.