Harris did well to win the first set courtesy of a tie break, but thereafter Rublev managed to get the better of his opponent and eventually win 6-7 (7/1), 6-4, 7-6 (7/5), 6-3.
It was a match that delivered plenty of drama in its two hours and 45 minutes, with Harris having a chance to win the third-set tie break, but ultimately falling foul of unforced errors to relinquish the momentum as Rublev closed the set out to take a 2-1 lead in the match.
Rublev would go on to break Harris to make it 5-3 - when Harris once again fired marginally too long - and from there was able to see off the South African, who had impressed in his opening-round victory over Zizou Bergs.
"The last game was a tough one but in general it was a super tough match," Rublev said.
"Lloyd played unbelievable and was serving really amazingly, so it was tough to do something. There were almost no chances on his serve.
"I was lucky that the toughest set was the third one and I was able to win that and it gave me a lot of confidence."
Rublev will play Adrian Mannarino in the next round after the Frenchman beat compatriot Valentin Royer 6-4, 6-4, 5-7, 7-6 (7/1).