Depends on what you mean by going anywhere, but it was a massive hit on the “northern soul” scene in England in the seventies and was bootlegged several times to fill popular demand. In fact, that is how Soft Cell knew about it.
Sort of the same thing happened to ‘The Only Way Is Up’ by Otis Clay, which was covered by “Yazz and the plastic population”.
And, maybe you don’t HAVE to, but you could also mention “Why Does the Sun Shine? (The Sun Is a Mass of Incandescent Gas)”, perhaps not their best known song but I’m sure there are lots of people who know it and don’t know it’s a cover.
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Georgia On My Mind**, 1930, by Hoagy Carmichael. Covered by everyone from Satchmo to Billie Holiday to Django Reinhardt. Also, Ray Charles.
In their early years, the Beatles displayed a knack for picking relatively obscure songs to cover, like “Twist and Shout”, “Dizzy Miss Lizzie”, “Money”, Slow Down", “Matchbox”, “Anna”, “Boys”, “Devil in Her Heart”, and “Mr. Moonlight”. I would guess that, for most people, their versions of these songs are the most well-known.
I have never heard either of those versions before, but I well remember Sandie Shaw’s huge British hit with the song (#1), also from 1964. Since neither of them wrote it, it is arguable whether Shaw’s of Johnson’s version should be considered a cover of the other, but as Naked Eyes were British, I am pretty confident that they thought they were covering Shaw, especially as she was already enjoying a bit of a career revival in the early '80s, being championed by the likes of The Pretenders and The Smiths.
“The Green Manalishi” was originally a Fleetwood Mac song, but the cover by Judas Priest is probably better known.
And Nazareth was mentioned earlier, “This Flight Tonight” was a Joni Mitchell song.
“Every Time You Go Away” a #1 hit for Paul Young, originally by Hall & Oates.
And speaking of Fleetwood Mac, “Somebody’s Gonna Get Their Head Kicked In Tonite” is a popular song with punk rock bands, but it was originally by the decidedly non-punk band “Earl Vince and the Valiants,” i.e. Fleetwood Mac.
Because it’s not. The OP is thinking of “Here Comes My Baby,” which is indeed a Cat Stevens original from 1967, although there was a very popular cover version by The Tremeloes that same year. The Drifters song “There Goes My Baby” was first recorded by them and later covered by many other artists, including Donna Summer.
Hall & Oates tear that song up on their VH-1 Behind the Music album. 
God bless you and them always!!!
Holly
Would “Wild Thing” count? It was written by Chip Taylor (brother of Jon Voight!), first recorded by The Wild Ones in 1965, and popularized by The Troggs a year later, but I’m not sure it was necessarily a “cover” of the Wild Ones version.
I’m not sure if “Woodstock” qualifies… it was written by Joni Mitchell and I believe her version was the first to be released, but the very differently arranged hit version by CSNY came very soon after.
Other way round. Deja Vu appeared before Ladies of the Canyon in 1970, but Mitchell performed it at least once in concert in 1969. I don’t when CSNY first performed it, though.
It’s a simultaneous release more than a cover. Much like “Wooden Ships” was produced in different versions by CSN and Jefferson Airplane. Neither was a cover of the other.
“Wild Night”-Originally a Van Morrison song, covered by John Mellencamp.
I agree with you, it’s really a simultaneous release.
I thought of that one too but apparently she co-wrote it, so I thought of hers as her version and not a cover, if that makes any sense!
The Isley Bros. “Work to Do”, covered in inimitable fashion by AWB.
I like the Quiet Riot versions too, but outdoing the originals? That’s just krazee talk.
I once had a very interesting talk with Rudy Isley about how he introduced “Twist and Shout” to John Lennon. The Isley Brothers did a British tour with the Beatles in 1962.
I damn near fell off my chair when I googled “Child in a Manager,” and found out “Morning is Broken” (which uses the same tune) is not a Cat Stevens original, having first been published in 1961.
The husband & wife team of singer/songwriter James Taylor and singer/songwriter Carly Simon made the definitive version of “Mockingbird.” Their version rocks.
Nobody’s mentioned “Turn, Turn, Turn”? The Byrds put out the best known version, but Pete Seeger wrote and recorded it first!!!
Blondie’s version of “The Tide Is High” is much better known than The Paragons’
Cheap Trick’s cover of “California Man” is better known than the original by The Move
Speaking of The Move, “Do Ya” was a bigger hit for ELO (although does it count as a cover when you rerecord your own song with another band?)
I don’t recall ever hearing Bowie’s original version of “The Man Who Sold The World” on the radio until after the Nirvana cover became a hit.