The meeting lasted 1 hour 23 minutes and ended with a score of 7: 5, 6: 2 in favor of the Russian tennis player, who is in eighth place in the ATP rating.
In the quarterfinals, Rublev will play either with Belgian David Goffin or Frenchman Jeremy Chardy.
Murray is a three-time Grand Slam winner as well as a two-time Olympic champion.
Earlier, Anastasia Pavlyuchenkova lost to Petra Kvitova in the second round of the WTA tournament in Doha.