County took the lead through Chris Zebroski's fine half-volley from 20 yards after 16 minutes and Michael Flynn's penalty doubled the advantage shortly after the break.
But the Dons hit back quickly when George Francomb's free-kick was deflected in by Kwesi Appiah before Francomb himself levelled up the scores with a deflected strike after 69 minutes
Newport sit in 13th place in the table after throwing away the two-goal lead but at least had the joy of welcoming back from injury Aaron O'Connor, who made his first competitive appearance off the bench since scoring in the play-off final last May.
And while admitting his side did themselves no favours by conceding straight after doubling their lead, Edinburgh was left with plenty to be happy about.
"It was as good as we've been for a long while in the first half and once we got that second goal with the penalty it was a schoolboy error to concede straight away," said Edinburgh.
"We all know that you're at your most vulnerable when you've just scored. We've been told that since we were eight years old, but we lost concentration.
"We should have gone on to win the game comfortably but I've got to take the positives because, as disappointing as the second half was, we were good in the first.
"We're delighted to see Aaron O'Connor getting half an hour. There's been some dark times for him. He's been in some bad places and had comebacks and setbacks but I'm really pleased for him and so is the whole group."
Edinburgh added: "Every game is a big game and we've got two home games left against teams chasing promotion so we have to give them a test."