The song, written by John Christopher, Mark James and Wayne Carson, was first made famous by Brenda Lee and Elvis Presley in 1972.
Johnny Cash's cover of Nine Inch Nails' Hurt came in second place, followed by The Stranglers' version of Dionne Warwick's Walk On By.
Jimi Hendrix's take on Bob Dylan's All Along The Watchtower came fourth.
Jeff Buckley's cover of Leonard Cohen's classic Hallelujah completed the top five.
The public voted for their favourites from a top 50 compiled by BBC producers and DJs including Fearne Cotton Trevor Nelson, Simon Mayo and Steve Lamacq.
BBC Radio 2 and 6 Music's head of music Jeff Smith said: "The Pet Shop Boys have an enviable catalogue of hits and there have been lots of great versions of this classic song.
"Whittling down the long list for the vote was tough but it's great to see that this track still resonates with people today."
The Pet Shop Boys reached number one in the UK and number four in the US with the track in 1987.
Willie Nelson also won a Grammy for his version in 1982.
The top 10 tracks in the vote are as follows:
The Pet Shop Boys' Always On My Mind has been voted the top cover version of all time in a BBC Music vote.