Kyle Traynor and Tim Visser were on the score sheet for the Scottish side while debutant Harry Leonard kicked two penalties but it was not enough for Bradley's side.
The defeat puts Edinburgh worryingly close to the foot of the table, just a point clear of their Italian conquerors.
And boss Bradley has warned his troops they need to buck their ideas up for next week's visit of Munster to Edinburgh.
"Credit to Aironi, they were the better side," said Bradley. "Our defence was poor and we gave away two soft scores in the first-half and were let down by an early score in the second which left us chasing the game.
"We scored two good tries when we played to our potential but will need to produce more of that if we are to challenge Munster at Murrayfield next weekend."
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