Cani opened the scoring for Villarreal just after three minutes following Joe Hart's deflection of Giuseppe Rossi's shot. While City took more shots before the break, it was an own goal by Carlos Marchena that finally equalised the scores.
Manchester City finally secured victory in the second half as Sergio Aguero struck home from close range.
Following their first win in the competition, the Blues are in third place with four points while pointless Villarreal are bottom.
Roberto Mancini's men will travel to Villarreal for the second tie on Wednesday 2, November.