I think the first game I have ever beaten was PC version of Michael Jackson's Moonwalker.
The game wasn't as good as SEGA's version (both arcade and Mega Drive) but it followed movie more accurately since we cnplay as Michael "Rabbit" Jackson from Speed Demon who appeared in the movie.

Well, I beat arcade version too but using continue with coins so I don't know if it's counted or not.
Months later after I have a Mega Drive, I finally beat Mega drive version too.
I think the first game I beaten was Alex Kidd in Miracle World.
It wasn't just the challenge of progressing through the levels, but the fact this was a time before save states and memory cards means 3-6 year old me had to beat the game in one sitting, which is a lot for a child, especially in Mum asks you to switch it off for dinner or bed time, or your older brothers wanting the TV and can just switch off the console if you refuse. But the challenge was also simply not getting Game Over and having to start the whole adventure again!

It's one thing to learn the patterns, enemy locations, traps and such, but when you break new grounds it's like "oh my god! I made it this far, but I have no idea what to do now!", and trial and error would just get frustrating the further in the game you were.
So, I'd like to say I beaten Alex Kidd... but i don't really remember doing so, I only know I got really far in the game over time through trial and error.
Not original enough.
He's rather original.