They are through to the last four after stunning four-time champions Toulouse with a 19-14 win at Murrayfield while Ulster Rugby won on the road when they beat Munster 22-16 at Thomond Park.
While Edinburgh are in uncharted territory for Ulster it ends a 13-year wait for another semi-final appearance and the chance to bring an end to constant references to 1999.
A prestigious place in the 17th Heineken Cup Final at Twickenham Stadium on Saturday, May 19 is at stake against the winner of the Bordeaux semi-final between defending champions Leinster Rugby and ASM Clermont Auvergne.
Former Springboks Ruan Pienaar, Pedrie Wannenburg and club captain Johann Muller have played major roles in Ulster's return to the prestigious semi-finals but young Irish stars like Craig Gilroy have also taken the chance to shine among the elite of European club rugby.
The goal-kicking form of Pienaar - 54 points so far this season - and Edinburgh captain Greig Laidlaw (82 points) could be decisive in such an evenly balanced contest with the breakdown also a critical area of the game.
Flanker Stephen Ferris will be in the thick of things there for Ulster while Edinburgh's Fijian No 8 Netani Talei picked up three Heineken Man of the Match awards in their Pool matches.
Bradley is in his first season as head coach of the Scottish capital city club and is adamant they will be firm underdogs at the home of Irish rugby although the teams know almost all there is to know about each other as annual Pro12 opponents.
On the European stage they have met four times in the Heineken Cup, the score there 2-1 to Ulster with one draw.
The one thing already assured is that a Celtic team will be at Twickenham for the showpiece final next month.