Any scene showing the five Bronze boys relaxing and doing things normal people do (eating, sleeping, exercising, playing guitar, playing soccer etc), mostly in early episodes of the Sanctuary chapter. Of special note is the scene where they're playing soccer with the kids of the Orphanage of Love.
Also the Rousing Speech that Hyoga gives to the poisoned and injured Shiryu and Seiya in the Temple of Scorpio, to give them their spirits back after Shun's nearly Heroic Sacrifice.
The anime-only scene where Ikki thanks little Helen for her support and carries her home in his arms.
Speaking of Ikki, the episode where we find out he's Back from the Dead includes a sweet scene where he wipes away Shun's Tender Tears and then is joined by all the other Saints as well as Saori, who are all crying happily.
Every single time that Seiya and Co. are able to defeat the villains (from the television series and the movies) by The Power of Friendship.
In the Hades chapter, Ikki refuses to give Shun to Pandora and stands up to her despite the fact that he can't be more than 6 years old at the time, doubles as a Moment of Awesome.
After Deathmask's defeat and death at the hands of Shiryu, the audience sees the latter "return" and find out that Shunrei, whom he believed to be dead thanks to Deathmask, had been saved. Additionally, he and Seiya realise that after Deathmask died, his victims have been released from their And I Must Scream situation. And also, in the anime the effects of healing water that Seiya once gave to Shiryu kick in, and Shiryu can see again.
The scene where Seiya saves Shiryu's life by kickstarting his heartbeat with a single punch is awesome, but crosses into this when Shun (who has been keeping Shiryu on his feet to help) tells Seiya that the apparently impossible task was actually Crazy Enough to Work. Cue to Jabu happily announcing that Shiryu's alive, to a huge street screen showing a "DRAGON, SAVED!" message while the crowds outside and inside cheer, Seiya gets everyone's congratulations while both Shunrei and Saoir cry Tears of Joy...