The Gunners have been rewarded with a quarter-final showdown with French side Toulouse at Murrayfield after their impressive home victory sent them top of Pool Two - one place above RaboDirect PRO12 rivals Cardiff Blues.
Edinburgh became the only Scottish side to reach the quarter-finals eight years ago, and Bradley thinks this year's achievement is a sign that the good times are ready to return north of the border.
"It's a huge step forward for Edinburgh Rugby," he said.
"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.
"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.
"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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