God Hand also qualifies, although the song itself actually plays during the ending credits. ("Lost a limb/in a fight/but don't worry babe, I'll be cool/the ultimate/power of a god/is now my secret tool")
Take a listen to "Still Alive", played at the end of the game Portal. This catchy, sweet-sounding, yet also slightly sinister song pretty-much sums up the plot of the game.
Similarly, the theme to the spoof Flash platformer You Have to Burn the Rope tells you exactly what you just did.
Then Portal 2 does the same thing with the song "Want You Gone". "So here we are again, it's always such a pleasure. Remember when you tried to kill me twice?"
As the credits role on Call of Duty 4, lead animator Mark Grigsby raps about the events of the game in character as Staff Sgt Griggs. He actually had a burgeoning rap career before joining Infinity Ward, so it doubles as The Cast Showoff.
Final Fantasy X for the song Otherworld. "An Other-world awaits you", "and you will fight, fight, fight... fight, fight, FIGHT", "Go now, into the sand and dust in the sky, go, for it is better to do than to die".
Persona 4 and its theme song for battles, Reach Out to the Truth, qualify for this trope. The entire game, for instance, is about seeing through the lies that cloud the truth from one's eyes, both self-imposed and those that cloud our view of others. Within the chorus, the lyrics include gems like "Oh god let me out / can you let me out / can you set me free from this dark inner world / Save me now, last beat in the soul". The majority of the game is about venturing into dungeons full of shadows - dark, repressed inner emotions given form - to save whichever character has been kidnapped that month.
Alan Wake does this in the song "The Poet and The Muse", which tells basically the whole plot of the game. The song itself is even a MacGuffin!