Jack Sock also won, beating Benjamin Becker 6-4, 6-3. Sock will play third-seeded Nicolas Almagro, a 6-2, 6-3 winner over Michael Russell.
On Wednesday, Sam Querrey and Dustin Brown set up a quarterfinal match. Brown beat top-seeded John Isner in three sets.
The 24-year-old Young has won consecutive matches on clay for the first time in an ATP World Tour event. Both Young and Sock are through to the quarterfinals for the first time on clay.
''Definitely the biggest win for me, (beating) a former top-10 guy who wins most of his matches on clay,'' Young said. ''Happy to move through, to be honest. I'm competing well and my shots are falling. I'm feeling good and confident.''
Monaco, the 2012 winner, said he was bothered by a sore shoulder but praised Young's play.
''He was strong today,'' Monaco said. ''He deserved to win.''
Monaco missed the last three months of the 2013 season because of a strained wrist.