There really is nothing like being dialed into a racing sim, especially when you're ahead of the pack.  Your whole world narrows down to the upcoming curve.  Go around enough times with a great car and you stop thinking, you just flow along while your mind makes calcuations and adjustments.  

Getting  to the Zeal Kingdom in Chrono Trigger for the first time. Zen, ethereal experience, with one of my favorite tracks in any video game ever.

I once got high on Belgian chronic and died 400 times in a row playing Super Mario Bros 2, does that count?

Honestly, I dunno if I can name any game where I feel this way. At some point in any game, I hit some sort of level of stress, whether it's fighting bosses or reaching the end of the game. There's just no relaxing in games.



..Well, maybe just walking around and exploring in Breath of the Wild? ...until ninjas or zombies appear xD

Not sure if this is Zen, but running in certain overworld areas and Fenmont in Xillia

Final Fantasy II (IV) SNES
Final dungeon on the moon was a big zen experience.
When my friend told me off-hand (no internet in early 90s) that there was an invisible
walk-way to discover the best sword in the game I felt so elated cruising that dungeon
and discovering the mighty Crystal sword after defeating the seething Dark Bahamut!

Journey

I'm taking this to mean 'in the zone', rather than just imparting calm through atmosphere.

For me it's hard to go past Bit Trip Runner 2 on the hardest difficulty.

Honourable mentions for Bayonetta 2, the music levels of Rayman and Splatoon during some of my finer online moments.

Flower, Silpheed, Rez, Time Crisis 1&2, Burnout 2, LttP and the original master system wonder boy.

When the credits roll after a fantastic game, and the tears nearly start to roll. I guess that is more emotional though. Still I feel the calm.