A music video was produced to promote the single, made especially for the European market, directed by Swedish-based director Matt Broadley.

[1] Larry Flick from Billboard magazine stated that this follow-up to the certified platinum "Another Night" "does not tamper with the European dance act's winning (and much-copied) formula of bouncy hi-NRG rhythms, topped with throaty male rapping and female chirping at the chorus.

Single has already begun to gather deserved airplay from a number of crossover and top 40 stations on import—its domestic release almost guarantees instant success.

"[2] Chuck Eddy from Entertainment Weekly wrote that the Berlin trio "has invaded U.S. radio by tap-dancing space-invader-disco synths beneath soul-diva testifying, "Sprockets"-accented raps, and Martian munchkin chatter.

[8] In his weekly UK chart commentary, James Masterton said, "All the elements that made "Another Night" such a smash are here once again, it may be a standard Eurohit formula but it works so why argue?

"[9] Pan-European magazine Music & Media encouraged, "Take a bit of this smooth slice of Eurodance, which serves as a preview for the forthcoming Space Invaders album.

[15] In Europe, "Run Away" entered the top 10 in Finland,[16] Ireland, and the United Kingdom, where it peaked at number six on 29 January 1995,[17] in its second week at the UK Singles Chart.

The song was a top-20 hit in Belgium and Sweden, as well as on the Eurochart Hot 100 and European Dance Radio Chart, where it reached number twelve[18] and twenty,[19] respectively in February.