They have never been there in such number. With five, including the first three seeds, the Celtic fringe, as some may have referred to it dismissively, now forms a majority large enough to raise the real prospect of the Celtic force monopolising the final.
If the only two unbeaten teams justify their top seeding, Leinster and Munster will get there.
The quarter-final rankings and the semi-final draw ensures that they will avoid knocking each other out as has happened twice, at the old Lansdowne Road in 2006 when Munster won 30-6 and at Croke Park three years later when Leinster settled the score, almost point for avenging point at 25-6.
In the event of both going the distance, Twickenham will be only the third collision between the neighbouring provinces since the Heineken Cup began in 1995.
Munster and Ulster have never met in Europe, a fact which adds to the historic nature of their shuddering set-to at Thomond Park over the Easter weekend in April.
They have only been in the last eight together once before, in the spring of 1999 when Munster fell foul of Colomiers in the quarter-finals and Ulster disposed of Stade Francais at Ravenhill en route to the final in Dublin where Colomiers found the occasion too much.
They have not been heard of as a force in France, never mind Europe, since.
Irish teams having got there in triplicate, the odds were always against them being kept apart until the semis.
Edinburgh's glorious transformation from perennial non-qualifiers to the surprise team of the tournament as the third best qualifiers ensured that there has never been a quarter-final line-up to equal it.
England arrived mob-handed with seven contenders among the 24 on the starting grid back in November only for six of them to find the competition too hot too handle.
Worse still, three of their past European champions have been hammered as never before at the hands of Irish opposition.
Munster's destruction of Northampton at Milton Keynes, leaving last year's finalists as stone-dead in European terms as the town's concrete cows, followed Leinster's half-century against Bath and Ulster's thumping of Leicester. The cumulative effect amounted to 144-70.
While Saracens are left to carry the Aviva Premiership flag on their Jack Jones as befitting their status of national champions, the French have fared only marginally better.
They have failed to secure a single home quarter-final, a fate which has befallen them just once before during 17 European seasons.
Their two survivors, Toulouse and Clermont Auvergne, first and second respectively in the Top 14, have lost the comfort of home territory and no team in Europe has been unbeaten at home for as long as Clermont.
The Michelin Men - 37 consecutive home matches since losing to Biarritz there in November 2009 - are away to Sarries.
Toulouse must go back to Murrayfield where they have not been since winning the 2005 extra-time final there against Stade Francais under Guy Noves.
Toulouse's venerable head coach, still by some distance the most successful the Heineken Cup has known, knows that Edinburgh are far from the relative push-over they were eight years ago when the then defending champions made light work of them (36-10) the last time the Scottish capital got this far.
"Edinburgh are a complete team," he says. "People think they have good backs but lack power in their forwards. I disagree. They have got better and better in the last few years."
Nobody will vouch for that more ruefully than the Cardiff Blues. They went into the climactic final round on Sunday knowing that a bonus-point win would guarantee them first place in the pool and the handsome reward of a home tie.
Instead it was Edinburgh under their Irish coach Michael Bradley who delivered the priceless fourth try two minutes from time against London Irish as touched down by a wing with a Welsh name.
Lee Jones is a Scot born and bred, just like every other member of Edinburgh's starting XV last Sunday with the notable exception of the Fijian flanker, Netani Telai.
Nobody in Wales ought to detract from Edinburgh's success because, when push came to shove, the Scottish capital outnumbered its Welsh counterpart in terms of support.
The pleading of Blues' skipper Paul Tito for home fans to crank up the noise against Racing Metro may not have fallen entirely on deaf ears but barely 8,000 turned up which left the Cardiff City Stadium two-thirds empty, alarmingly so given what was riding on the result.
Almost 11,000 poured into Murrayfield, a second successive record crowd for Edinburgh following last month's 13,246 against Glasgow in the RaboDirect Pro 12.
A challenging scenario was made all the more so by the late withdrawal through injury of Chris Paterson and David Denton.
Edinburgh took that, and London Irish, in their stride.
Once they had put the Exiles away, the Scots could still not be sure whether they had done enough to win a home tie or the short-straw of a quarter-final against Leinster in Dublin.
They and their fans had to go through the emotional wringer of watching Xavier Rush fall within feet of scoring, then Richie Rees within inches before the final blast from referee Andrew Small signalled that Edinburgh had made it to the promised land of the quarter-finals.
Cardiff will have to worry about Leinster instead, a prospect which explained why they ended up feeling a deeper shade of blue than their strip.
Edinburgh's adventure has the potential to run beyond the quarters and nobody ought to be the least bit surprised if it does.
A semi-final against Munster or Ulster in Dublin is more than a possibility, all the more so given Edinburgh's booming box-office popularity. Scotland now has a team to shout about and nobody appreciates that more than Edinburgh's goalkicking captain, Greig Laidlaw.
"The noise the fans created as we piled on the pressure was awesome," he said. "It's fantastic to make it this far but we are not finished yet…"
Fantastic, too, for the RaboDirect Pro 12 with more teams from Ireland, Scotland and Wales to be found in Europe's top eight than ever before.
Best XV picked from the quarter-finalists:
15 Rob Kearney (Leinster)
14 Andrew Trimble (Ulster)
13 Casey Laulala (Cardiff Blues)
12 Wesley Fofana (Clermont Auvergne)
11 Timoci Matanavou (Toulouse)
10 Ronan O'Gara (Munster)
9 Ruan Pienaar (Ulster)
1 Rhys Gill (Saracens)
2 Ross Ford (Edinburgh)
3 John Afoa (Ulster)
4 Leo Cullen (Leinster)
5 Paul O'Connell (Munster)
6 Stephen Ferris (Ulster)
7 Netani Talei (Edinburgh)
8 Louis Picamoles (Toulouse)
63x46.png){kind=logo}