Animes about gaming such as New Game!, No Game No Life, and Gamers! don't count. Neither do games based on manga/anime.
One example that counts is Pokémon, often looked at as an anime that came before its video games. But it's incorrect. It's the other way around.
The 3rd Fire Emblem and other pre-FE7 games stayed in Japan, but the anime OVA was released overseas, but Marth in dub version was referred to as "Mars". Usually "s" is used in place of "th". Like "thunder" being "sunder". This OVA didn't become popular, but Super Smash Bros. Melee popularized the FE series.
Steins;Gate was also looked at as originally an anime, but actually it's an anime based on the game of the same name.
The first example that always comes to mind is normally anime based on Visual Novels with the Fate series being probably one of the best examples.