The Premier League champions, without their World Cup stars, were beaten by the Scottish side, with centre-back Dedryck Boyata suffering a particularly chastening match
After looking bright in the early stages, City - without their World Cup stars - conceded a 27th-minute penalty when Dedryck Boyata bundled Gary Harkins to the ground.
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The Scottish Championship winners continued to press and were rewarded seven minutes before the break, with the unfortunate Boyata at the centre of the action once again. Trialist Luka Tankulic climbed well to head goalwards but it was a heavy deflection from the Belgian centre-back that diverted the ball beyond Caballero.
City boss Manuel Pellegrini elected to make nine changes at the break - including the withdrawal of new signing Fernando - and the Premier League side could have pulled a goal back two minutes into the second period.
Substitute Bruno Zuculini got his head to a cross from the right but the Argentine failed to get enough purchase on his close-range effort.
City rarely looked like turning the result around, despite the introduction of the likes of Alvaro Negredo and Samir Nasri, and Pellegrini will no doubt seek improvement when they face Hearts on Friday.