The song was recorded on April 18, 1983 and released as a single in December 1983, with "Union Sundown" as its B-Side.
"Sweetheart Like You" peaked at #55 on the Billboard Hot 100 on January 28, 1984, staying on the chart for a total of nine weeks.
The song also appeared on the ARIA Charts in Australia, reaching #74.
In addition to Dylan, the song features Mick Taylor, Mark Knopfler who plays the lead guitar solo, Alan Clark on keyboard, Robbie Shakespeare on bass, and Sly Dunbar on drums.
Rolling Stone listed the song at #75 on its list of 100 Greatest Bob Dylan Songs.