Victories in the two opening rounds - against defending champions Toulouse and not forgetting a try bonus point at Neath-Swansea Ospreys - have given Todd Blackadder's team the perfect platform to go where no Scottish team has gone before.
Heineken Cup newcomers Leeds Tykes travel to Meadowbank on Friday night still very much in the Pool 2 hunt themselves, with Edinburgh coach Frank Hadden declaring: "Both us will be thinking we can qualify from the group and that it will largely depend on these two games. Edinburgh and Leeds will be of like minds and that should make for a fascinating game.
"I know quite a lot about Leeds Tykes, I have seen them a few times this season, and Phil Davies can be justifiably proud of the progress they have made over the last few years.
"Considering the very impressive quality of the opposition it is a massive task for us and it is extremely tough to qualify for the last eight in this tournament.
"But we know we have put ourselves in a position to go through and that is our sole aim over the next month - to be the first Scottish side to qualify for the Heineken Cup quarter-finals."
Edinburgh have already suffered the agony of losing in a cup final, beaten 27-21 by Ulster in the Celtic Cup final a few weeks ago, but it was not all bad news.
"We were very disappointed - particularly when we saw the wet weather in which the game was going to have to be played," said Hadden. "But full credit to Ulster, they are an outstanding side and on the day deserved their victory while we were always chasing the game.
"You do not get many of these sort of chances and, although the opportunity was there, we did not take it.
"However, it was another opportunity for Chris Paterson, who is very inexperienced at stand-off, to play against a world class No 10 in David Humphreys and I am sure he will have gained valuable experience from the occasion.
"The national selectors are encouraging us to play Chris there and we have to give it a crack. If he can keep learning at his present rate he will turn into a fine stand-off, he has all the bits and pieces."
And while no Scottish team has made the last eight of Europe's premier tournament, Hadden defends that position.
"Considering our limited playing resources, and things like the lack of real high intensity matches for our junior rugby, I don't think we have done badly in Europe, I certainly don't think we have under-achieved," he said."I am getting a lot of pleasure out of watching the style of game we are producing - Scottish rugby has not been renowned for its passing skills and that is what we have been working on."