Roared on by his home crowd, the 27-year-old won the first set 7-4 in a tiebreak before breaking his big-serving opponent in the first and ninth games of the second.
The local favourite, who lives in nearby Santa Monica and was a student on the UCLA campus hosting this week's event, will meet Czech Radek Stepanek in the last eight.
"I honestly put Rob's serve up there with (Ivo) Karlovic, (Wesley) Moodie and (Andy) Roddick, all the best servers that are out there," Fleishman told reporters after sealing victory in a little under an hour and a quarter.
"Against a player like Rob you go into the match knowing you're going to get probably one or two chances the entire time.
"It almost feels like he can hit a service winner any time he wants. He was holding a lot easier than me but I knew if I just stayed in there and played my service points as well as I could, I would get an opportunity.
BIGGEST UPSET
"I think I proved that this is an official level that I can play at, instead of just one good outing," added Fleishman, who produced the tournament's biggest upset by knocking out top-seeded Fernando Gonzalez of Chile in the previous round.
"This is the first time I have ever reached the quarter-finals of a Tour event and I hope to take it further."
Wildcard Fleishman, who qualified for the tournament by winning an eight-player shootout last week, paid tribute to the fans at the Los Angeles Tennis Center for lifting his spirits.
"When I hear a few cheers from the crowd, it gives me a little bit of energy and I feel much better."
Fleishman, whose world ranking is 162nd, believes his shot-making and mental strength have improved considerably this season.
"I have calmed my nerves better this week than at any other tournament I have played and I think that has been a huge factor so far in my stringing a couple of good matches together.
"And I wasn't nervous today, I was so focused on every single shot."
