When starting online you are presented with two options - Quick Game and Lobby.
Although Quick Game races are specified in the achievement description, the quick game option isn't actually required for the achievement.
(At the time of writing this, the Quick Game option actually failed to connect every time for me).
Select Lobby, Create, choose a game mode and set visibility to Private. You can then invite a friend (or another profile if you have a second console) and trade wins.
The second player doesn't need to actually do anything. They can sit at the starting line and then just abandon race from the menu once you're done. However, if they want to work towards the 25 completions in each event, they should finish the race.
Some game modes - V-Rally Cross and Buggy - have AI opponents.
Just win three races and you'll earn this achievement.