The Best Is Yet to Come from Metal Gear Solid is a good one, although I can't claim to understand the language:

Gaelic I think. 

One of the greatest songs in video game history from The Ninja Warriors arcade game. Still counts even if the lyrics are synthetic voice.

Really? 3 pages in and have not seen this one yet!

Granted, lyrics are in an invented language, but still, lyrics.

It's up there, just not as an embed.

Bowie's song New Angels of Promise for Omikron: Nomad Soul, later released (with lyrics changed in chorus) on his album Hours in '99

Bit of a cheat, since it's traditional Bulgarian song, but can't skip "...Steel for Humans" from Witcher 3:

Baba Yetu from CIV IV:

Was about to post Wind Nocturne. 

Now some people on this forum might be too young for this but this song became a meme and viral with Shii Cat flash-animation and later Long Cat video. This song along being a nice song became really famous on early internet and I guess a lot of people have heard it but don't know where it comes from. 

Most of the Katamari Damacy OST applies, and it's fantastic.

Both come from a PS2 mech game called S.LA.I.