In order to play minigames, you must have either a second local account signed in, or be playing against an online opponent. At the Start Game screen, make sure MULTIPLAYER MINIGAMES is set to "ON." You will then get to play one minigame between each round, for a total of two minigames per game.
This achievement simply requires you to win both minigames in a single full game. If your minigame happens to be Correct Order, you don't have to correctly place all 4 items to win. You only need to get more correct than your opponent.
Note that minigames will NOT be in the game if you're playing alone or against only AI opponents, even if you have the option set to "ON."