Roger Miller was the first popular singer to put the song on the charts. Bobby McGee was a woman in his version. Written by Kristopherson and first covered by Roger Miller, a kind of off-beat county singer. Before Kris and before Janis, he put the song into popular music.
1963 by New Order might qualify for this. It starts off with the lyrics
“It was January 1963 when Johnny came home with a gift for me”
Now, that alone doesn’t mean it is being sung from the point of view of a woman, but later on it goes
“Johnny came home with another wife”.
Apparently, looking online, it’s a rather weird fantasy song about JFK hiring Oswald as a hit man to kill his wife so he can be with Marilyn Monroe (which makes no sense anyway since she died the year before JFK), but he hit him instead. But I think the narrator is a woman??
Betty by Taylor Swift is sung from the perspective of James, who is a boy according to Swift, but that’s interpreted differently by others.
Cindi Lauper covered Prince’s When You Were Mine without changing the lyrics:
I know (I know )
That you’re going with another guy
I don’t care (Don’t care )
'Cause I love you, baby, that’s no lie
I love you more than I did when you were mine
I always assumed Beloved Wife by Natalie Merchant was sung from the perspective of a husband, but I guess that’s not necessarily true.
My favorite version of The Band Played Waltzing Matilda is by June Tabor. The story of an Australian soldier - Gallipoli and all that. The first thing she sings is “When I was a young man…”
j
Just about my fave, but the Animals did a good job also. They also sang “boy” IIRC.
Good one. I’m pretty sure it is.
Pearl Jam had two on VS: “Daughter” and “Elderly Woman Behind the Counter in a Small Town”
No because it’s this:
The first line is “The way I see it, he said, you just can’t win it”
The song is Joni quoting David Geffen. The performer is a woman, and the character is a woman describing what a man said to her.
Reading the lyrics again, you’re correct. I now have to re-think this song. 
Not that I’m aware.
That wasnt a good way to say it, sorry, altho certainly Eric sang a bunch of stuff after the Animals, like with War.
RICHARD SHINDELL has a song called “Money for Floods” where he narrates as a woman.
“Weird Al” Yankovic’s “Truck Drivin’ Song” qualifies, assuming the narrator is a deep-voiced woman and not a man in drag.
“Luka” by Suzanne Vega
Any male singer’s version of “I’m Just Wild About Harry.”
“Hands Clean” by Alanis Morissette, at least the verses.
The Arctic Monkeys covered Girls Aloud’s Love Machine and didn’t bother to change the lyrics so that should qualify.
Ladies, you’re damn right
You can’t read a man’s mind
We’re living in two tribes
And heading for war
Well, nobody’s perfect
We all gotta work it
But fellas, we’re worth it
So don’t break the law
Sweet Guy by Paul Kelly is a jewel in the Paul Kelly discology:
I went to town with a moody man
A handsome Dr. Jekyll
He was right by my side turning into Mr. Hyde
I ran for cover but I ran too slow
I was stitched by strangers
And they shook their heads
That somebody could do the things he did
One thing I’ll never understand
It’s become my problem
And it’s something that’s right out of my hands
My hands are clean
What makes such a sweet guy turn so mean?
But when his band performs with the magnificent vocals of Vika Bull the song effectively sexually reorientates, and it is a barn stormer.