Devvarman and Harrison overcame an early-evening rain delay that lasted 2 hours, 30 minutes.
Harrison kept his composure despite double-faulting five times in the first set -- seven overall -- and made the best of a third-set argument with Belgium's Xavier Malisse to upset the No. 4 seed 6-7 (3), 6-4, 6-4.
Devvarman, who won all 21 of his first-serve points, beat Japan's Tatsuma Ito 6-1, 6-3 and will play Fish in the quarterfinals.
Harrison will face Ram, a 7-5, 2-6, 6-2 winner over Australia's Lleyton Hewitt.
Fish, who is ranked No. 9 in the world and won this event last year at the Atlanta Athletic Club, hasn't played outdoors since losing to Rafael Nadal in the quarterfinals at Wimbledon.
"It's the first match back on outdoor hard courts, and you always feel a little pressure coming back where you've had success, so I'm happy to just move on," Fish said. "It's only going to get tougher."
Fish improved to 3-0 against Mahut. They hadn't played since the 2007 Australian Open.
"He's a tricky opponent because he serves well and he plays very aggressively," Fish said. "I got down a break in that second set and broke him right back to snatch the momentum back from him."
Harrison, who improved to 5-10 this year and 10-20 in his ATP Tour career, said he and Malisse argued midway through the third set. After Harrison broke Malisse to make it 2-3, they exchanged words during the changeover.
"When I broke him back, he said some things to me that I wasn't too happy he said, so I said some things back to him that I wasn't too happy he said," Harrison said. "There's no reason to go into real detail about it because whenever you're in the heat of the moment, whenever you're competitive and you both want to win. We've always been fine with each other. We've never been close, but we've always been OK with each other."
Harrison won the last three games, breaking Malisse's serve in a ninth game that included a double-fault.
"Whenever someone makes a threatening comment to you in the match, you know, the ultimate goal ... is to get the last laugh in the situation," Harrison said. "I was going to make him play a heck of a match if he was going to be that guy."