Idah pounced on a fumble by Gers keeper Jack Butland in the last minute of normal time to fire in the winner.
City were hit with two goals in nine first-half minutes by their city rivals.
Hernan Crespo's Al Ain steamrollered 10-man Yokohama F-Marinos 5-1 in the Asian Champions League final return leg to lift the trophy for the second time in their history on Saturday.Al Ain, the oil-rich United Arab Emirates' most successful club, won the inaugural Champions League final in 2003 and were runners-up in 2005 and 2016.