I usually had one of these awe-inspiring gaming experiences per generation:
NES - Super Mario Brother 3, the game just blew me away with its technical achievement, length, fun and variety. I actually got this game BEFORE I got an NES.
SNES - Link to the Past. I remember seeing previews for this game in Nintendo Power and getting super hyped for it. Then it came out and just pulled me into the world. It remains my favorite video game of all time
N64 - Super Mario 64. I first played this game at a kiosk in a game store and was amazed. I hadn't been following the 64 development too much, so it was kind of like a game bomb dropped on me. OoT was amazing as well, but that answer is cliche, and it didn't have quite the impact of that first glorious SM64 experience.
PS2 - GTA 3. This game solidified my love for open world games, and I still think it remains as one of the best GTA games made. I have probably 100% it 5 or 6 times and would happily go back and do it again.
Gamecube - Metroid Prime. This game was unbelievable. I was skeptical because I despise FPS in general, but this game was so much more.
Wii - Wii sports, because in a growing world of repetitive, mundane game play, it introduced a new gameplay experience for a long time gamer.
Wii U - Lego City Undercover. And odd choice but I just loved everything about this game. It was so charming, fun and a truly excellent and fresh take on the open world genre, which has jumped the shark a bit as of late.
EDIT: It is kinda funny that you used the term "magic" as that is exactly how I described my first experience with Mario 64 to my girlfriend who had never played it. She is (some what strugleingly) playing through it now
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