Even if you're the most hardcorextreme gam0r to ever walk the planet, these scenes will make you want to reach out and hug your console. No other medium can so effectively force a viewer to empathize. Keep tissues handy.
Series and games with their own pages:
In Knights of the Old Republic, Mission Vao and Zaalbar have their Crowning Moment of Heartwarming when it's revealed you are Revan. This trope can be subverted in a nasty fashion if you reclaim your former identity and butcher them both.
Most of the light side dialogue after defeating Bastila on the main deck of the Star Forge also qualifies.
If playing as the male protagonist you are allowed the option of bringing back your fallen Jedi companion from The Dark Side, at the pivotal moment of a battle for control of the galaxy, the final confrontation with Darth Bastila on the Star Forge. The most effective method is not by reciting the Jedi Code or making a generic appeal to come back to the Light, but simply by admitting to her that you love her... and asking her to admit to herself that she still loves you. Given that this editor first played KOTOR right after suffering through the whole idiotic 'Jedi cannot be allowed to love!' dogma from "Attack of the Clones", words cannot describe the satisfaction of that moment.
At the Rakata Temple, after the Rakata insist the protagonist has to enter the temple alone, suddenly Jolee Bindo and Juhani appear and steadfastly declare their intention to come with the player character because they are her friends. This editor cheered.
At the lightside ending, Malak and Revan reconcile as Malak lays dying, Revan asks for forgiveness for leading Malak down The Dark Side, Malak tells him he's not to blame, for it's Malak who decided to stay down that path. Malak then says he wishes he had been able to cast the dark side aside like Revan did, and comments that "In the end, I am nothing."
With a light-side female character, you get to watch Carth let go of his issues and grief, reunite with his son, and allow himself to love again.
In Knights of the Old Republic II, after you finally defeat Kreia, she allows you to ask her questions about the future. When you ask about your companions, she tells you that they become the basis for the renewed Jedi order, after all the Jedi and all the Dark Jedi have died, save for Kreia (and she's dying herself). Doubly heartwarming because Kreia hates living with the Force, and by dying she'll be free. Had the game been completed when it was released, the game would have had a Kill 'em All ending. Thank god for those meddling executives.
Well, except all the awesome stuff they cut out, like any real closure for any of the charecters('cept Kreia) extra missions and general polishing, I agree with you. A Killem All ending would have sucked.
In II, you get a chance to go back through the most torturous, nightmarish parts of the Mandalorian War that scarred you for life, and do it right this time, saving your comrades from certain death. Their ragged cheer is all the more heartrending if you think that this may give peace to their souls (especially when you see the actual battlefield).
Also, the romantic dialogues. All of them, really, but this troper actually teared up during the Sky/Male PC interlude.
Wild Flower's good ending seems like a Crowning Moment to this troper, Chai Ka leaves her body and grants her the life she never had.
This troper considers the Open Palm ending of the drowned orphans quest to qualify and then some. the cutscene where Kindly Yushan is burying their bones as beautiful music plays, and then Bin and Miao bow to you before dissipating is just immensely heartwarming.
For that matter, the Open Palm endings of most characters qualify. Particularly the male protagonist's if he hooked up with Silk Fox (Dawn Star's is also good).
The nice resolutions to the Tien's Landing sidequests always make this troper smile. Notable are the incidents with Ai Ling and Zhong the Ox Carrier. Especially Zhong.
An older example. In Infocom's sequel to Planetfall, Stationfall, at one point you and your Robot Buddy, Floyd, come across a young, "unborn" robot named Oliver. Upon seeing it, Floyd has this to say:
"Oliver is a nice name. I hope Floyd is around when Oliver wakes up. Floyd remembers waking up for the first time. It is scariest and loneliest time for a robot. If we be here when Oliver wakes up, Floyd would like to teach Oliver to play paddleball. Paddleball helped Floyd forget how lonely and scared he was until he found friends...like you."
Hyperdimension Neptunia Victory has a great one as long as you're on the path to get the "Good" and "True" Endings. At the end of Chapter 8 Neptune, Plutia, IF and Compa do everything in their power to restore a brainwashed Peashy's memories. They show her the dolls, remind her of how much fun they used to have...and of course, Nep's Pudding. The scene ends with Peashy remembering them and she and Neptune hug.
Seiken Densetsu III gets a few, though only if you have the required character in your party.
Angela hugging her mother and crying, just happy that her mom is back to normal and loves her.
Kevin being reunited with Karl, finding out that Karl was never dead.