I see that a bunch of people have mentioned the Ace Attorney games, but I'm not sure if "emotional" is really the right word there. They affect me quite a bit, but somehow the thrilling aspect seems to be quite a bit more important than the "emotion".
The thrilling aspect is more important than emotion, I agree, but don't say that taking down the culprit in the last case of Trials and Tribulations isn't very emotional.
Almost every single last case in any Phoenix Wright game is pretty sad. You don't get a truly happy ending, but instead, you end up re-opening past wounds and learning the bitter background of everyone involved. The "special case" of the first Phoenix Wright (The one where you're with Emma, not Maya) had Edgeworth resigning at the very end, while a bunch of characters such as the cowboy and Emma's sister are jailed. Even though everyone cheers in the end, well, it's not that happy.