The win firmly upset the form book as Edinburgh lie second from bottom in the RaboDirect PRO12 table while Toulouse top the French Top 14.
And Bradley believed it was the partisan support which gave his players the edge as they fought fire with fire against the illustrious French club.
"We had close on 40,000 people here and I think they were a factor," said Bradley.
"With 20 minutes to go there was one score in it and the boys really dug deep with the crowd behind them.
"In the final 10 minutes, Toulouse were really pushing the button to try and get some turnover ball but our lads took the hits and supported each other well.
"On the second yellow card, when Ross [Rennie] was binned, we were delighted that they went for a kick at goal.
"I think the reason they did that was because it was such a tight game.
"If they had put the pressure on and got a potential try it could have caused us difficulty.
"To hold them to that score in that period was great credit to the boys and the effort they put in."
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