Manchester City failed to reach their first Champions League final after an uninspiring semi-final second-leg display at 10-time winners Real Madrid.
Fernandinho clipped the post in a rare City chance, but Real dominated as Bale hit the bar and Joe Hart saved from Luka Modric and Cristiano Ronaldo.
Real saw out the win to meet Atletico Madrid in the final later this month.
The English side knew they potentially only needed one away goal to progress, but appeared reluctant to go for broke in the final stages - despite being encouraged to attack by the vocal 4,500 away supporters.
The home side's only other moment of concern came in the closing minutes when Sergio Aguero's speculative effort flew on to the roof of the goal.
Now they will meet Atletico at Milan's San Siro on 28 May, in a repeat of the 2014 final which Real won 4-1 after extra-time.