Cozart's line drive to center field off Sam Demel plated fellow Reds prospects Chris Heisey and Yonder Alonso for an 9-8 edge before Carlos Rivero (Indians) capped the comeback with a two-run double.
Heisey went 3-for-4 with three RBIs and scored the tying run, two innings after after Phoenix scored eight times to take an 8-5 lead.
"It was a definitely an up-and-down game. When they had that rally, I said, 'Here we go again,'" Heisey said. "We had a chance to come back, it was a lot of fun, to be honest."
Heisey has driven in at least one run in five of six AFL games and ranks second in the league with 15 hits. He said he told his teammates in the eighth inning that a comeback was brewing.
"We've been running into some bad luck so far. We knew [Phoenix] had come back multiple times already, I'm thinking, 'We cant catch a break,'" explained Heisey, who is riding a six-game hitting streak. "It was tough to watch.
"I felt for our pitcher. He made some good pitches, gave up a few dribblers. They hit a couple balls hard, but the wind was blowing way out. Balls scrapped over the wall -- it was agonizing."
The Desert Dogs felt some agony themselves in the ninth when Heisey's bases-loaded single plated Jose Vallejo to spark the rally. Brandon Hicks was thrown out at the plate on Alonso's base hit, but Cozart followed with a two-run double for the lead.
"It was a lot of fun," Heisey said. "At some point, we were kinda joking in the outfield during a pitching change. I said, 'We need to break up the sad streak and we have to do it ourselves. It's not going to happen on its own.' So it was awesome anytime you can be part of a comeback like that."
Rivero finished 4-for-4 with two RBIs for Peoria, which opened the season with six straight losses. Astros Minor Leaguer Chia-Jen Lo earned the win, striking out two over the final two frames.
Heisey, who batted .314 with 22 homers and 77 RBIs at two Minor League levels this season, said he's enjoyed playing alongside Alonso and Cozart.
"We get a long really well; all the guys on the team get along," he said. "It's awesome to meet new guys, but it's nice to have those guys that you're great friends with. We've been able to hang out, eat, go out together."
The Messiah College product, batting .522 in Arizona, said he came to the Fall League without any big expectations in terms of production.
"I kinda had a rough second half," admitted Heisey, who hit .347 at Double-A Carolina before batting .278 in 63 games with Triple-A Louisville. "I wanted to come down here and get back on track. So far, I'm feeling pretty good. Been able to have some good at-bats, work on some things, especially my timing.
"I've been getting some pitches to hit, I'm feeling good, timing is good, and that's all it takes to get on a hot streak. Get some balls up in the zone. I'm taking some good swings so far."
Athletics prospect Grant Desme hit a go-ahead three-run homer off Carlton Smith (Indians) in the Desert Dogs' eight-run seventh. Blue Jays farmhand Brad Emaus followed with a two-run shot to right off Dan Meszaros (Astros) to cap the outburst.
Rays right-hander Matt Gorgen (0-1) took the loss after surrendering five runs on five hits and a walk while retiring one batter in the ninth.
Danny Wild is an editor for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.