Lacy scored with 1:31 to go for Green Bay's first lead since the first quarter and after the Packers trailed 26-3 at halftime.
"It took me everything not to cry," said Packers coach Mike McCarthy, whose team won its second straight without Rodgers after going winless in the first five, including the Chicago game when the star QB got hurt. "I'm just drained. Just the sheer emotion. It was incredible."
Playing on the same field where Rodgers was the MVP when they won the 2011 Super Bowl, the Packers matched the 1982 team in a September game against the Los Angeles Rams for the largest rally.
The Packers (7-6-1) kept their playoff hopes alive with Rodgers possibly returning next week. He was close to medical clearance but missed his sixth straight game with a broken collarbone. He was wearing on a headset on Green Bay's sideline.
"Most guys would pack it in 'Guys, it's not our season. Let's go home and get ready for next year,'" said Green Bay cornerback Tramon Williams, who had an interception overturned on review before a game-clinching pick that was ruled incomplete and reversed on replay. "It's not that vibe here."
The vibe in Dallas could be negative for a while, with Dez Bryant walking off the field before the game ended and refusing to talk to reporters in the locker room, simply shaking his head at them after finishing with 153 yards and a touchdown.
"This is one of the hardest losses that I've experienced," Cowboys owner Jerry Jones said. "That's a shame that we've lost that ballgame."
The Cowboys (7-7) blew a great chance to pull even with Philadelphia in the NFC East but still control their playoff fate. They finish with Washington and the Eagles.
"We still control what we feel like we can get done," Romo said. "It just doesn't feel like it right now because we let it slip away."
With Dallas leading 36-31, Bryant made a lunging first down after a catch just before the 2-minute warning that looked as if it would Dallas help hold off a huge Green Bay rally. But Romo threw behind Miles Austin after escaping a sack and Sam Shields intercepted at midfield.
Monday night football
P Baltimore at Detroit6:40 p.m. TV • ESPN