The 28-year-old Monaco improved to 9-2 on clay, and reached his third semifinal this year. He won on the clay in Vina del Mar in February and lost to top-ranked Novak Djokovic in the semifinals in Miami last month.
Monaco broke Anderson's serve to take a 6-5 lead in the second set, and Anderson fought off two match points before Monaco finished him off.
The 33-year-old Russell, a qualifier, converted 39 of 55 first serves to earn his first career semifinal berth.
Second-seeded John Isner defeated Ryan Sweeting 7-6 (1), 7-6 (4) in the evening quarterfinals, and Feliciano Lopez advanced when Carlos Berlocq retired with a leg injury. Lopez won the first set 6-4 and the players split the first four games of the second before Berlocq had to stop.
Monaco finally feels he's regained his 2008 form, when he reached No. 14 in the rankings. He sprained his left ankle late that year, then was sidelined three months in 2010 with a left wrist injury.
Monaco came into the tournament ranked 16th and has won 14 of 19 matches this season.
"I'm trying to not think about that," Monaco said. "I'm just trying to improve my tennis. If I'm winning matches, then the ranking and all the points, that will come along."
The 6-foot-8 Anderson had six aces, but Monaco won two pivotal break points.
"I knew it was going to be a very close match," Monaco said. "You never know when you play big-serving guys. I was focused all during the match, and I tried to win my serves, and then when I got a few opportunities for a break, I played my best tennis there. I tried to be very, very focused on those moments and I did it."
Russell was appearing in his first career quarterfinal on clay.
"I feel good," Russell said. "Obviously, I would've liked to maybe win that in straight sets. I felt like I let up a little bit in the second. I was playing really well in the first set, but I'm happy to get through it."
Harrison was seeking his first career title. The Austin resident was also appearing in his first career quarterfinal on clay. He slipped to 12-10 this year, though his last four losses had all come against players ranked in the top 20 in the world before Friday's defeat.