The song was first performed, and recorded as a demo, by the band Cap'n Swing, which featured Ocasek, Orr, and guitarist Elliot Easton as members.
In this early version, the recurring keyboard theme between the verse lyrics was significantly different.
[2][3] Rolling Stone critic Kit Rachlis said in his review of The Cars that "the songs bristle and -- in their harsher, more angular moments ('Bye Bye Love,' 'Don't Cha Stop') -- bray.
"[4] Jaime Welton, author of 1001 Albums You Must Hear Before You Die, described the track as a "fan favorite", praising Elliot Easton as an "unsung hero, littering songs like 'Bye Bye Love' with staggeringly good fills.
"[4] AllMusic critic Greg Prato called it one of the "lesser-known compositions [that] are just as exhilarating" as the "familiar hits" on The Cars.