The defending World Series champions trailed 6-1 after the fourth, with starter Clay Buchholz walking a career-high eight in only three-plus innings.
But Ortiz tied it by hitting a three-run homer off Ervin Santana in the fifth. With Red Sox fans at Turner Field chanting "Papi! Papi!" the World Series MVP then gave Boston the lead with a bases-loaded sacrifice fly off Ian Thomas (1-2) in the seventh.
Following pregame declarations of confidence from manager John Farrell and general manager Ben Cherington, the Red Sox came back to end their worst skid since an 11-game slide in 1994.
The losing streak was the longest for a defending World Series champion since the depleted 1998 Marlins had two 11-game skids.
Justin Upton drove in three runs with two hits for Atlanta.
Edward Mujica (2-1) got the win with a scoreless sixth inning following a 1½-hour rain delay.
Chris Johnson singled with one out in the ninth before Koji Uehara capped his 10th save with Andrelton Simmons' game-ending double-play grounder. Uehara was Boston's seventh pitcher.
Buchholz began the day with an unsightly 6.32 ERA, but even with his poor start he hadn't walked more than three batters in a game. He couldn't overcome unprecedented control problems, as he allowed six runs and four hits.
Buchholz was wild from the start. He walked leadoff hitter Jason Heyward and catcher A.J. Pierzynski visited after a first-pitch ball to the next hitter, B.J. Upton.
Pitching coach Juan Nieves made his first walk to the mound when Buchholz walked Freddie Freeman.
Buchholz matched his season high of three walks in the second inning and added three walks in the Braves' three-run third. Justin Upton had a two-run double and Gerald Laird had a run-scoring single off Buchholz's left foot.
Buchholz didn't get out of Atlanta's three-run fourth. Heyward walked and scored on B.J. Upton's double, and Justin Upton and Simmons added RBI singles.
As Boston's early deficit grew, the frustration from its players became more obvious. After his groundout in the fourth, star Dustin Pedroia stalked around the dugout, shouting in anger before slamming his hand on the bench. He seemed to be upset about a ball-strike call earlier in the at-bat.
The Red Sox rallied with five runs in the fifth against Santana. Pedroia drove in two runs with a single before Ortiz hit his tying homer to left-center.
It marked the third straight start Santana allowed at least five earned runs. He gave up five runs on six hits and three walks in five innings.
Following the rain delay, Thomas walked two batters following an infield hit by Brock Holt, loading the bases. Ortiz drove in Holt with a fly ball to right field, giving Boston a 7-6 lead. Pierzynski's single up the middle pushed the lead to two runs.
Heyward led off the eighth with a single to center field off Andrew Miller but slid past second base trying for a double. He was tagged by Pedroia while reaching back for the bag.
NOTES: The crowd was a sellout of 48,501. Most fans remained during the rain delay. ... Buchholz singled in the second inning for his second hit in the majors. ... Holt had two hits and scored twice. ... The Red Sox recalled C-1B Ryan Lavarnway from Triple-A Pawtucket and optioned RHP Alex Wilson to Pawtucket. Lavarnway had a pinch-hit groundout in the seventh. ... Ortiz made his first start of the season at first base. The team placed 1B Mike Napoli (sprained finger) on the 15-day DL on Sunday. ... Braves LHP Jonny Venters, trying to return from his second Tommy John surgery, had more discomfort throwing from the mound before the game. ... Braves RHP Aaron Harang will face Red Sox LHP Jon Lester on Tuesday night.
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