County goalkeeper Elliot Parish allowed a Ryan Tafazolli header through his grasp to gift Stags a 28th-minute lead, only for home stopper Alan Marriott to spill the ball at the feet of Rene Howe to bag a 50th-minute equaliser.
But Anthony Howell proved to be Mansfield's hero, leaving the Exiles with just one win in 12.
Edinburgh said: "That is very difficult to take. We should have dealt with it better.
"We dominated that second half. But we have got to make sure when you don't win it, you don't lose it.
"We have now got to show a quick response.
"When we had a spell after the goal we should have been more clinical.
"We bossed it and restricted them to very little on their home pitch. But we then paid the price for not dealing with a second phase ball.
"We are lacking that bit of belief. They were there for the taking but we didn't capitalise on that.
"It's been out story since Christmas and only ourselves can turn it around.
"We have lost momentum - it's as simple as that and we have to deal with it. We know there are reasons. You can't just turn it on and off when you feel like it.
"We need to start showing a bit of consistency as we can't keep relying on teams below us. We need to stand up and be counted as a group.
"Thankfully the games are coming thick and fast so we can't dwell on this result too long."