Playing in front of 37,881 spectators, a record crowd for a quarter-final in Britain, Edinburgh had an early try by Mike Blair and a string of successful kicks from Greig Laidlaw to thank for the historic 19-14 success as they became the first Scottish side to reach the last four of the Heineken Cup.
The home side had to come from behind to overhaul the visitors, who scored an outstanding try through Timoci Matanavou and added nine points from Lionel Beauxis.
Bradley said: "We have the ability to trouble anybody on our day and there was plenty of confidence in the side.
"We've played a lot of good rugby in this competition, and under pressure we've been able to deliver in critical moments in matches in the pool stages.
"And I think we rode our luck a little bit in it as well. Today was very similar to that and we were on the right side of the result."
Toulouse coach Guy Noves was magnanimous in defeat, saying: "We were expecting them to be a dynamic team ready to give its best.
"It wasn't a surprise that Edinburgh won and they deserved to win because they played well.
"We produced a good first half and a very bad second half, so we have to accept that it's our fault and congratulate Edinburgh."
Edinburgh were reduced to 13 men at one point as Allan Jacobsen and Ross Rennie were sent to the sin-bin in quick succession, and Bradley added: "Going down to 13 men was not wise.
"You could lose a game quite comfortably if you are down to 13 men for 10 minutes. We could easily have lost the game then."
However, that indiscipline did not ultimately prove costly and Bradley was delighted to secure a semi-final spot, preferably against his native Munster, who face Ulster on Sunday.
"I'm a Munsterman so I hope it's Munster," Bradley added.
"I had a lot of good wishes from people all over Ireland for today's match.
"All of this is about Edinburgh Rugby and, by extension, Scottish rugby. The fact that Edinburgh Rugby are now in the semi-final of a European Cup is a fantastic achievement for the club and gives us an opportunity to move forward.
"We have a very strong connection with the fans who came today and it's up to us to convert that into Pro12 attendances and try to move the club up."
Speaking about this game, Bradley added: "We were obviously very fearful of Toulouse's ability, in particular up front, but we also felt that just to play Toulouse one way would be too easy for them to defend.
"So what we tried to do was use multiple tactics, sometimes going wide, sometimes going close. We kicked more ball today than we would normally kick because Toulouse's defence is very strong and they score a lot of tries from turnovers.
"Their breakdown work is excellent, so we just had to keep varying the target."
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