Ram, who won 32 of 35 first-serve points, knocked off 20-year-old Grigor Dimitrov of Bulgaria, the No. 5 seed, 6-4, 6-4.
Australia's Matosevic closed his 6-7 (3), 6-2, 7-5 win over No. 6 seed Igor Kunitsyn of Russia with two aces.
Lleyton Hewitt of Australia ignored recurring pain in his foot to beat Phillip Simmonds 6-4, 6-4; Yen-Hsun Lu of Taiwan held off Matthias Bachinger of Germany 7-5, 7-6 (4); and Ryan Harrison took a 6-1, 7-6 (5) victory after Japan's Yuichi Sugita won only five of 39 first-return points.
It marked Ginepri's first ATP appearance since the right-hander broke his left elbow in a biking accident last September. Ginepri, who lives in nearby Kennesaw, spent months rehabbing.
"I felt a little passive in the beginning, started off a little bit nervous," Ginepri said. "I wasn't moving to the ball and wasn't hitting the ball as well as I needed to. It took a little time to get used to the rhythm and the court."
Matosevic, the No. 1 seed in the Atlanta qualifying tournament, and Ram, the sixth qualifying seed, are accustomed to playing their way into ATP events.
For Matosevic, the victory was his first on the ATP Tour in seven events this season and his second in 11 career matches.
"The match should've been easy," Matosevic said. "I was 5-3 up, 30-0 serving for the first, and he came up with a few good points, but I think the difference was being aggressive out there because it was so hot. I was finishing the points quicker than he was."
Matosevic was pleased to end his day with two blistering serves.
"Yeah, I served for the 5-4, 30-0 and I missed a routine backhand short one, and he came up with two good points to break me," Matosevic said. "I just kept on fighting. I broke back and then served it out to love with two aces. Thank God."
For Ram, qualifying has become a way of life, too. Tuesday's victory marked his first winning match in an ATP event 250 or higher since last October in St. Petersburg, Russia.
Thick heat in the afternoon and early evening caused some high bounces on the hard courts of the Racquet Club of the South. Hewitt was among those who needed a few games to feel comfortable on the surface.
"It took me halfway through the first set to adjust to that, but I felt like I returned fairly well from start to finish today," Hewitt said. "I didn't give (Simmonds) too many cheap points on his service game, and his biggest weapon is his first serve. So I felt like any time he missed his first serve I was really on top of him for his second serve."
Hewitt, who missed nearly four months after undergoing foot surgery earlier this year, is still playing with considerable soreness, but at least he no longer needs painkillers or shots to reduce inflammation.
"It's still a day-to-day thing, and I'm still battling it a bit," Hewitt said. "Obviously I haven't played with as many injections and painkillers over the last couple months. I was trying to do everything possible and get through a few matches this week and take it one week at a time."
Ginepri overcame nerves by staying patient as Haas used a tough backhand chip to make him play some tough shots. Toward the end, Ginepri felt the game slowing down, and his service game boosted his confidence.
"I served my way out of trouble a few times," Ginepri said. "It's a lot easier to hit aces and big serves than playing 10 to 15 shots to win the point. Hopefully I can do that throughout the week."
Mahut will play No. 1 seed and defending champion Mardy Fish on Thursday.