Ocarina of Time has some of these. The first one you usually come across is the treasure chest game in the town. It's a simple enough game: each room contains two chests. One contains money (increasing as you progress through the game), the other contains the key to the next room. You're only allowed to open one chest in each room. It's purely luck-based, unless you cheat and use the Eye of Truth to look inside the chest. On my current playthrough of Ocarina of Time, I got all the way to the end room on the first attempt, without using the Eye, and picked up the heart piece at the end.
The Minish Cap also has one of these, except it's more of a gamble. One chest rewards you double your bet (initially 10 rupees), the other gives you nothing, and after each pair you can choose to double your bet and try again. The first time I ever played it I won all the way up to the maximum bet on the first attempt. I've never managed to repeat that feat since.
Back in Ocarina of Time, there's another reportedly luck-based challenge: getting the heart piece that's buried in Kakariko Graveyard. I was using a guide to pick up the last few heart pieces, and according to it you need to watch the boy and note where he pretends to dig, then go back at night and one of those places will have the heart piece. He pretends to dig in about a dozen different spots, so I very carefully sketched a map and noted all the places to try. Guess what came out of the first one I got Dampé to dig at?
I managed a similar trick on some skill-based ones as well. As young Link I got all ten targets in the shooting gallery on the first attempt in this playthrough (which makes up for the hours my sister and I spent trying to achieve this when we first player Ocarina of Time, all those years ago), and as adult Link it took three attempts. Then there's the Gerudo Horseback Archery, which I've never tried before. So I decided to have a go, thinking that it'd take at least one attempt to get my eye in, and managed to score exactly the amount needed for the heart piece on the first attempt. Go me!