Here are some rap songs and other related music which i feel led up to the popularity of rap and mainstream acceptance. These were among the most obvious rap tracks of the 80s, even if they were bubbling in the underground. They were songs that most high school mix tape makers like myself knew about and blasted from their car radios and boom boxes.
Shawn Brown- Rappin Duke (1985)
Sugarhill Gang-Rapper's Delight (1979-80_
Newcleus- Jam on it (1983)
Twilight 22- Electric Kingdom (1982)
Soulsonic Force (aka Afrika Bambaataa and the SOulsonic FOrce)- Planet ROck (1982)
NWA- Boyz n the hood, Gangsta Gangsta, Radio (1987-88)
AfroRican- give it all you got (1987)
2 live crew- Throw the dick, Get it girl, WE want some action(edited)(1986-87)
Dana Dane- Cinderfella , Nightmares (87)
Salt N pepa- Push It, Tramp, Chick on the side(love bandit) (86)
Fat Boys-wipeout, crushin, the twist (87)
Joeski Love- Pee Wee's Dance (86)
World Class Wreckin' Krew featuring Dr. Dre- Juice (85)
Egyptian Lover- Egypt Egypt (1984)
Herbie Hancock-Rockit (although not rap, its sound made industrial noises acceptable to the mainstream, which rap later hedged on) (1983)
Rob Base and DJ EZ ROck- It takes Two, Get on the dance floor (1988)
Blondie- Rapture (a pop song, but Debbie was one of the first to get away with rapping on a pop record)
Teena Marie- Square Biz (r&b singer but also rapped on some of her tracks like this one and her most popular "Lovergirl")(1984)
Chaka Khan- I feel for you (the male rap part in this song is actually by Grandmaster Melle Mel) (1984)
Grandmaster Flash, Melle Mel, and the Furious Five- WHite Lines (1982) (later covered by Duran Duran)
Frankie Smith- Double Dutch Bus (1980) This roller skating rink anthem was goofy, but paved the way for future generations to come.
Herbie  Hancock Vs. Grandmixer Dst - Megamix (based on the song Rockit' this song became quite an anthem of its own with other Hancock tracks such as overdrive, rough, future shock , and others mixed in with Rockit, the basis of the track, Dst's turntable tricks just barely float over the top of the mix, its breakbeat heaven) (b side of Rockit 45 single and twelve inch single) (1984)
Of course, if you wanta know who were the first artists to really slam home with multi-platinum success it would be into the 90s before a rap artist would achieve having more than one album go platinum.  We all know The Chronic by Dr. Dre was what broke the west coast into things, and ultimately into the MTV spotlight. If you look at everything, it was a buildup to this, that took a long time, but it happened, for better or worse.

I'll add some.  In addition to djunlimited's list here were a few tracks that seemed to make it in quite a few of my early mixtapes that paved the way:

Boogie Boys - "A Fly Girl" (1985)
L.A. Dream Team - "The Dream Team Is In The House!" (1986)
U.T.F.O. - "Roxanne Roxanne"/"The Real Roxanne (With Roxanne Shante) (1984)
KC Flightt - "Planet E" (1989)
D.J. Chuck Chillout & Kool Chip - "Rhythm Is The Master" (1989)
Kurtis Blow - "The Breaks" (1980)
RUN-D.M.C. - "Walk This Way", "You Be Illin'", "It's Tricky" (1986)
Gucci Crew II - "Sally That Girl" (1989)
Kyper - "What Gets Your Body Hyped (XTC)" (1989)

Rapture - Blondie (1980)
Basketball - Kurtis Blow (1985)
Amityville (The House on the Hill) - Lovebug Starski (1986)
Let Your Backbone Slide - Maestro Fresh Wes (1989)