David Bowie - Jump and the long version of Jazzin’ for Blue Jean.
Bjork - Joga
Cibo Matto - Sugar Water
More to come.
The song’s arranged more or less identically, sure, but I don’t think Roth’s voice is particularly like Prima’s - Prima’s got a fairly distinctive voice, and Roth doesn’t really catch the same sound. (Also, in the Just a Gigolo portion, he refers to himself by name - ‘Just a gigolo, everywhere I go, people know the part Dave’s playin’’. Think he changes up the rhythm of the scatted line in I Ain’t Got Nobody, too, but I can’t find the studio version of Prima’s, which I haven’t heard in years, just a live performance.)
Van Halen - Hot For Teacher featuring Phil Hartman as the voice of Waldo.
Huh? What? Really? Cool trivia find! I’ve always found that to be the finest combination of video/band/song ever made.
More cool ones…
Thomas Dolby, Blinded Me With Science
Madness, One Step Beyond
Wall of Voodoo, Mexican Radio
Pink Floyd, Learning to Fly
Jethro Tull, Kissing Willie
Here you go. You’re right: it’s really nothing like it (especially since Keely Smith is singing with him) other than similar phrasing. Roth’s version is very good.
More:
[ul]
[li]“Buffalo Stance” - Neneh Cherry (I’ve always loved the chintzy, yet vibrant, look of this clip.)[/li][li]“Where the Streets Have No Name” - U2 (Bono in full Beatles/Jesus mode.)[/li][li]“When We Was Fab” - George Harrison[/li][li]“Bittersweet Symphony” - The Verve [/li][li]“Domino Dancing” - Pet Shop Boys (Is this the gayest video of all time?)[/li][li]“Crucified” - Army of Lovers (Or is this?)[/li][li]“Black Hole Sun” - Soundgarden[/li][li]“One” (long version) - Metallica[/li][li]“Hurt” - Johnny Cash (Greatest video ever? Um, yeah.)[/li][/ul]
Very likely more to come.
My list:
NiN-“The Perfect Drug”
NiN-“Closer”
Duran Duran–their entire catalog, but I have a place in my heart for the long version of “New Moon On Monday”
Def Leppard–“Hysteria” and “Heaven Is”
Rush-“Distant Early Warning,” “Half The World”
There’s more but that’s the ones I can think of at the moment.
My top two favorites (Muse’s “Knights of Cydonia” and A-Ha’s “Take on me”) are already posted, so here’s a few others:
Supreme Beings of Leisure - Never the Same
K-OS - Crabbuckit
Flobots - Handlebars
Sigur Ros - Hoppipolla
Muse - Bliss
Within Temptation - Memories
Yes - Owner of a Lonely Heart
Apocalyptica - I Don’t Care
You reminded me of one of my favourites, Headrush - “The Man I Used To Be” by k-os. I especially like the dance scene on the sidewalk between the two strangers.
I am also very fond of Rihanna’s "Umbrella - so many gorgeous visual elements.
Yah, I love all the great dancing K-os throws into his vids. (Your k-os link is broken, btw, but it’s easy enough to find).
Thought of a few more, and not sure how I forgot them in the first list!
Birthday Massacre - Looking Glass
Tool - Sober
Tool - Prison Sex
Adding some of my own while trying not to duplicate anybody else’s pics:
80’s
New Order - True Faith
Phil Collins & Philip Bailey - Easy Lover
Eddie Money - We Should be Sleeping
Tom Petty And The Heartbreakers - Jammin’ Me
Fleetwood Mac - Big Love
90’s
Orgy - Stitches
Texas - In Our Lifetime
REM - At My Most Beautiful
Morcheeba - Rome Wasn’t Built in a Day
I’m not a fan of hip-hop but I do like this song and the video is cool
Lauryn Hill - Everything is Everything
00’s
Gorillaz - Dare
Keane - Is it Any Wonder?
Remy Zero - Save Me
X-Ecutioners - It’s Goin’ down
Lifehouse - Halfway Gone
A really old fave from Denmark:
Bakerman by Laid Back.
Everyone in the band is a skydiver, and Lars von Trier filmed them performing the song after jumping out of a plane.
Another:
“Imitation of Life” - R.E.M.
I thought about mentioning that one because the nearly 4 minute video is just 20 seconds of footage repeated over and over again.
I had never seen the Cash video before, nor heard that song. It’s a very powerful self-indictment. I was shocked to see how much he looked like my older brother, who is now 74. A dead ringer.
Huh - that’s odd - still works fine for me. Maybe it’s pulling from a cache. Yeah, it’s easy enough to find. 
For some reason Nirvana’s “Smells Like Teen Spirit” had a big impact on me. I think it was the tattooed cheerleaders.
Desert - Emilie Simon
Libertine - Mylene Farmer (NSFW - nudity)
Human Behaviour - Bjork
The “Dreams” video by Van Halen that featured the Blue Angels.