Bradley spent seven seasons in Galway with Connacht before making way for current boss and former assistant Eric Elwood two years ago.
In the battle of the two former Irish internationals on Friday, Gregor Hunter's try proved the difference for the Scots in a 19-14 win, and Bradley admitted it was a relief to get a first competitive victory since his move to the Scottish capital.
"Obviously I know them very well over in Connacht after the amount of time I stayed there," said Bradley who won 40 caps at scrum-half for Ireland.
"But it's far more important to me to get that first win under our belts.
"We didn't play as well as we would have liked against Cardiff Blues in the first game, and having lost the second as well it was a really important win for us.
"We've seen an improvement in each of the games and we need to keep that going against Aironi on Friday.
"We know they will be very physical and that is something we are going to have make sure we cope with.
"They are in a similar position to where we were last week without a win, so we know that doesn't mean a great deal."
While Edinburgh have had to deal with the loss of eight players to the Rugby World Cup, the omission of versatile half-back Greig Laidlaw from the Scotland squad has been a huge boost for Bradley.
The Irishman commended Laidlaw's professional attitude at overcoming his international disappointment while also revealing his excitement at seeing the next generation of Gunners coming through the ranks.
"Greig is playing very well at the moment. He obviously came back very disappointed after missing out on the World Cup," he added.
"It's not easy with a World Cup coming up, for the two years before everyone thinks about it and then when you get into the outer squad you think you are really in with a chance.
"It was difficult for Andy Robinson and in fairness to the three guys who have come back to us (Laidlaw, Ben Cairns and David Denton), they've all come back in and worked hard.
"I always enjoyed this period when I was with Connacht. It's a chance to see completely different sides and see what the future holds.
"We've given some youngsters the opportunity so far this season and they have really impressed so that's great going forward."
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