The track, which is officially released on July 30, serves as the latest single from her new album Boys Don't Cry.
The song was originally written and recorded by Hall & Oates and reached number four in the US charts on its release in 1976.
'Sara Smile' follows 'P.F Sloan' from the covers record, which entered at number three on the UK album chart earlier this month.
Rumer recently attended an exclusive concert honouring Burt Bacharach and Hal David hosted by Barack and Michelle Obama at the White House, where she admitted she bought a £6.99 tin of biscuits as a gift.
Watch a live performance of 'Sara Smile' below: