Anderson, who also made the last eight in 2015, will face either Sam Querrey of the United States or Mischa Zverev of Germany for a place in the semi-finals.
"I really imposed my game at the start," said Anderson. "But Paolo never gives up, he runs for everything.
"I had to dig really deep."
Anderson went into the match not having been broken in 43 service games but that run came to an end in the second set. DM