Curtis, inducted into the Rock and Roll Hall of Fame as a member of The Crickets in 2012, died on Friday, his wife of more than a half-century Louise Curtis confirmed to the Associated Press.
His daughter, Sarah Curtis, wrote on his Facebook page that he had been suddenly ill.
Curtis wrote or co-wrote hundreds of songs, from Keith Whitley's country smash I'm No Stranger to the Rain to the Everly Brothers' Walk Right Back, a personal favourite Curtis completed while in army basic training.
Bing Crosby, Glen Campbell, Bruce Springsteen and the Grateful Dead were among other artists who covered his work.
Born during the Great Depression to cotton farmers outside of Meadow, Texas, Curtis was a childhood friend of Buddy Holly's and an active musician in the formative years of rock, whether jamming on guitar with Holly in the mid-1950s or opening for Elvis Presley when Elvis was still a regional act.
Curtis' songwriting touch also soon emerged: before he turned 20, he had written the hit Someday for Webb Pierce and Rock Around With Ollie Vee for Holly.
Curtis had left Holly's group, the Crickets, before Holly became a major star.
But he returned after Holly died in a plane crash in 1959 and he was featured the following year on the album In Style with the Crickets which included I Fought the Law - dashed off in a single afternoon, according to Curtis, who would say he had no direct inspiration for the song - and the Jerry Allison collaboration More Than I Can Say, a hit for Bobby Vee and later for Leo Sayer.
Meanwhile, it took until 1966 for I Fought the Law and its now-immortal refrain "I fought the law - and the law won" to catch on: the Texas-based Bobby Fuller Four made it a Top 10 song.
Over the following decades, it was covered by dozens of artists, from punk (the Clash) to country (Johnny Cash, Nanci Griffith) to Springsteen, Tom Petty and other mainstream rock stars.
"It's my most important copyright," Curtis told The Tennessean in 2014.
Curtis' other signature song was as uplifting as I Fought the Law was resigned.
In 1970, he was writing commercial jingles when he came up with the theme for a new CBS sitcom starring Moore as a single woman hired as a TV producer in Minneapolis.
He called the song Love is All Around, and used a smooth melody to eventually serve up lyrics as indelible as any in television history:
"Who can turn the world on with her smile? / Who can take a nothing day, and suddenly make it all seem worthwhile? / Well it's you girl, and you should know it / With each glance and every little movement you show it."
The song's endurance was sealed by the images it was heard over, especially Moore's triumphant toss of her hat as Curtis proclaims, "You're going to make it after all".
In tribute, other artists began recording it, including Sammy Davis Jr, Joan Jett and the Blackhearts and Minnesota's Hüsker Dü.
A commercial release featuring Curtis came out in 1980 and was a modest success, peaking at No. 29 on Billboard's country chart.
