I found another cover song that falls into the veering-into-better-than-the-original territory.
That song is A Case of You - an already near-perfect Joni Mitchell song reinterpreted by the amazing chanteuse k. d. lang. If a closer listen to the vocals doesn't make you weak - the piano certainly will.
Not to put too much focus on just one song - the entire album Hymns of the 49th Parallel , an homage to fellow Canadian singer/song writers, is buried treasure (originally released in 2004) and it's absolutely flawless.
To say this is on heavy rotation at my house is a HUGE understatement. I could drink a case of you and I would still be on my feet...still be on my feet...