Edinburgh produced some scintillating rugby to sweep aside a London Irish side that offered little in an attacking sense at Murrayfield as Bradley's men secured top spot in Pool Two just ahead of Cardiff Blues, who failed to secure a bonus point in their victory over Racing Metro.
The last and only time a Scottish side made the last eight was Edinburgh's appearance in 2004, and their reward for reaching the quarter-finals this year is a home showdown with French powerhouses Toulouse.
Bradley said: "It's a huge step forward for Edinburgh Rugby.
"There is a talented group of players here. They made a commitment to each other in August or September and set out some objectives on a way to play the game, and a commitment to certain areas of the game that we weren't so good at in the past.
"The boys have done what they set out to do and from that point of view it's great for Edinburgh Rugby."
The Scottish outfit's passage was secured by tries from Tim Visser, Jim Thompson, Netani Talei and Lee Jones, plus a successful afternoon with the boot for captain Greig Laidlaw, who landed all four conversions and a brace of penalties.
The Exiles, who had no chance of progressing from Pool Two, posed a limited offensive threat. Their points came from a Jonathan Joseph try and two penalties by Adrian Jarvis.
Bradley, whose side led 20-6 at the break, added: "I thought we played that game pretty well. We controlled the game as best we could and we controlled the possession we had.
"We were, relatively speaking, comfortable with the way the first half went and we just said more of the same in the second half. It got a bit rocky, with a couple of turnovers. But the boys were fantastic near the end."
Bradley was also pleased with the amount of spectators as a record Murrayfield Heineken Cup crowd of almost 11,000 turned up to watch the match, adding: "It's great to see the size of the crowd. Week on week it's building."
Blocking Edinburgh's passage to the last four is the formidable form of four-time Heineken Cup winners Toulouse in April.
Bradley said of their next opponents: "Toulouse at home is much better than Toulouse away. They can be stunning in attack. But if you do your work properly and you are clinical, you can put them under pressure."
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