The original of that song was by Barrett Strong, not the Beatles.
The Rolling Stones recorded a superb cover of “Just My Imagination” by the Temptations on their Some Girls album.
The Georgia Satellites recorded the best-known version of Chan Romero’s “Hippy Hippy Shake.”
I may have the definitive example. (Link goes directly to sound file.)
My #1 vote is always Wilson Pickett’s “Hey Jude.” He took a #1 song by a #1 group and made it even better. He owns that song.
Madeleine Peyroux’s “Dance Me to the End of Love” is superior to Leonard Cohen’s version.
Quiet Riot’s “Come on Feel the Noize” was better than Slade’s original.
And, I always felt that Clapton’s unplugged version of Layla was an improvement over his original. (Of course, I’m not sure that can officially be called a cover.)
ETA: Oh, and Eva Cassidy’s cover of Fields of Gold did what I hadn’t thought possible: make a Sting solo song appealing. (Actually, her covers were usually at least the equal of the originals.)
I kind of agree and kind of disagree with the OP. 
This really gets into the heart of “covers”. See, you get a great cover when a musician makes it his own. Too often in pop music you’ll hear a cover, but it’ll be someone just imitating the original. Big whoop. But when a great musician or group takes a song (whether well known or not) and makes it their own and really really get into it, then the result is a wonderful new thing that you can really enjoy.
Of course, just because someone does a great cover – and Johnny Cash doing “Hurt” fucking rocked, man – it does not take away from the original, IMHO. While in my mind Hendrix’s version of “All Along the Watchtower” is possibly the greatest single of all time, Dylan’s original version is still interesting.
Anyway, I think when someone does a great cover, it speaks well for both the songwriting skills of the author and the musical skills of the performer.
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Nah, Derek and the Dominos version of Layla is much better. How can you let that piano part at the end off?
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Listen to Eva sing “Fever.”
I love pretty much every cover that Cake ever did, but especially “Perhaps, Perhaps, Perhaps” which was originally recorded entirely in Spanish and then in English.
I’m a big fan of Iron and Wine’s cover of “Such Great Heights,” originally done by The Postal Service.
Indigo Girls did a couple of great covers that I love, one a straightforward cover of “Down By The River” that I like as much as the Neil Young version, and one of “Romeo and Juliet” that completely blows away the Dire Straits original.
Joan Baez covering Bob Dylan’s…well, everything and anything. Especially her live version of “Lily, Rosemary & the Jack of Hearts”
Great pick. IIRC, Trent even told a reporter that it wasn’t his song anymore. He may have wrote it first, but Johnny Cash took it away from him and Trent had no regrets about it.
I refrained from mentioning it up to now, but I think Sinead O’Connor’s cover of “House of the Rising Sun” fits this description to a T. It’s not better than that covered by the Animals, but she makes it a completely different experience for me, with her haunting Irish lilt making chills run up and down my spine. “Poor, poor boy . . .”
Carpenters’ cover of “Calling Occupants of Interplanetary Craft” beats the Klaatu original, mainly because of Karen’s vocal.
I’ll see your “Flaming Lips” doing “I Can’t Get You Out of My Head” and raise you one Alanis Morissette’s cover of “My Humps”
(I don’t think the Flaming Lips used enough cowbell in that cover.)
My husband’s contributions to better covers are:
Prince’s “Every Day Is A Winding Road” (original by Sheryl Crow)
Elvis Presley’s “Blue Suede Shoes” (original by Carl Perkins)
Vanilla Fudge’s “You Keep Me Hanging On” (original by The Supremes)
Duran Duran’s “White Lines” (original by Grandmaster Flash)
I came in here to say just that, but you beat me to it. Klaatu’s vocal is just a vocal, but Karen’s vocal makes that song soar.
At least I can offer Judy Collins’ version of Leonard Cohen’s “Suzanne.” Another where the cover artist’s voice makes the song work–Cohen wrote a beautiful song, but his voice just wasn’t what that song needed. Judy, on the other hand, made it work very well.
Cake did a great version of I will Survive, Foo Fighters also did my favorite version of Stairway to Heaven
The Dickies’ blazing uptempo version of “Nights In White Satin” is far superior to the Moody Blues’ original.
I wouldn’t say any covers of Bo Diddley’s “Who Do You Love” are better than his recording, but it’s interesting that both Tom Rush and George Thorogood did very different but extremely good versions (Thorogood recorded an alternate to his widely played release that smokes).
Los Lobos did a nice sleazy version of Buddy Holly’s “Midnight Shift” (a low down song to begin with - it’s hard to believe Mr. Clean actually wrote it).
*If Annie puts her hair up on her head
Paints them lips up bright bright red
Wears that dress that fits real tight
Starts stayin’ out late till the middle of the night
Says that her friends give her a lift
Well Annie’s been working on the midnight shift
*
Speaking of Dylan covers, Manfred Mann followed up “Blinded By The Light” with “You Angel You” which was also an improvement.
Not on my planet. In fact, I’d nominate Chuck Berry as one performer whom no one singing his songs has ever been able to top.
The ELO version of “Roll OVer Beethoven” is fun, but for one thing it goes on too long.
The original Sea of Love is just vapid poppy la-la. The Honeydripper’s version, w/ the orchestral background, is absolutely TRANSCENDENT.
What about those covers that completely change the song, to the point where it’s barely recognizable? I’m not sure we could even count those here. That said, I for one find AC’s version of “Stayin’ Alive” far superior to the Bee Gees, but that’s just me.
The Ramones’ version of Tom Waits’ “I Don’t Wanna Grow Up” may not be better, but it’s truly one where they make it their own. Where the original is full of despair and angst, the cover is downright defiant. They could easily have called it “I Ain’t Gonna Grow Up!”
Johnny Cash and Joe Strummer’s cover of “Redemption Song” is truly amazing. It’s also impossible (for me, at least) to separate it from the fact that it’s one of the last things either of them recorded.
It wasn’t so much a cover as it was a tribute one-time performance, but I thought Korn kicked royal ASS playing Metallica’s One, and seeing Lars rocking out to it just seals the deal.
S^G