A sublime touchdown pass from P.J. Walker to Moore looked to have stolen victory for the Panthers, tying the game with only 12 seconds left.
But the extra point that would have won it for the Panthers was pushed back by 15 yards after Moore ripped off his helmet in celebration and earned an unsportsmanlike conduct penalty.
Eddy Pineiro's attempt from 48 yards missed, and the Falcons won 37-34 in overtime after another Pineiro miss.
Moore accepted his mistake, even if he described the celebration as "a natural reaction", as the Panthers slumped to 2-6, now two games back on the 4-4 Falcons in the wide-open NFC South.
Steve Wilks, the team's interim head coach, said: "We didn't find a way to finish. We had too many opportunities to win this football game, and we didn't find a way to get it done. And that falls on me.
"We've got to make sure we're smart enough. Something to learn from. Celebration penalty, big play like that, we've got to keep our poise.
"It was a great job with D.J. coming up with the big play in the end zone. Great pass by P.J. But as a team we've got to make sure that we're smart and just ready. Take it to the next level, next step."
Wilks added: "It didn't come down to that [penalty]. Of course, it would have given us an opportunity to win, but offense and defense, special teams, we could have had it way before then.
"And with me, I'm encouraging [Moore] and trying to build him up. I told him, 'Hopefully we give you the opportunity to come back and win it for us,' which unfortunately we didn't."
Moore had been a hero for the Panthers the previous week as he caught the opening TD pass from Walker in a shock defeat of the Tampa Bay Buccaneers.
"D.J. is smart enough to understand that's something to learn from," Wilks said.
"Very talented player, glad he's on our team. But moving forward that's something we'll learn from, and we won't make that same mistake."