The Welsh side were made to work all the way by Aironi who picked up four tries, two from Nick Williams, while Chris Czekaj touched down twice for the visitors.
The Blues had crashed to a demoralising 34-13 defeat to Glasgow Warriors a week ago and were clearly intent on putting that game behind them - securing a bonus-point win.
Aironi meanwhile were looking to pick up a second successive home win, having downed Edinburgh at the Stadio Zaffanella a fortnight ago, and, despite picking up a bonus point themselves, will look back with regret on the 11 points they missed from the kicking tee.
The home side had made the perfect start as winger Sinoti Sinoti grabbed the game's opening try after just two minutes to stun the visiting Blues supporters.
But Cardiff bounced back immediately through the returning Dafydd Hewitt, the inside centre tying things up just three minutes later with Sweeney's conversion putting the side from the Welsh capital in front 7-5
Sweeney added a penalty to consolidate the lead before fly-half Naas Olivier, who was so impressive in the win over Edinburgh, landed his first kick of the game midway through the half.
Sweeney notched up another penalty five minutes later and Welsh international full-back Czekaj ran in the Blues' second try of the game, Sweeney's conversion making it 20-8.
Olivier, who came back into the team having been relegated to the bench in last week's loss to Leinster, missed his second shot at goal as he pushed a penalty wide on the half-hour.
The South African redeemed himself somewhat knocking over a penalty just before the break as the Italians went in trailing 20-11 at half-time.
Aironi looked to have got themselves back into it ten minutes into the second half as former Munster No.8 Williams crashed over but Olivier again failed to add the extras.
And as they had done in the first half, the Blues' reaction was immediate with Williams' opposite number and fellow Kiwi Xavier Rush crossing the whitewash, Sweeney's conversion making it 27-16.
However the home side refused to bow down and centre Gilberto Pavan pulled them right back into it after 63 minutes, but Tito Tebaldi, taking over from the misfiring Olivier, was also off-target with his conversion attempt.
Another penalty from Sweeney looked to have made it safe at 30-21 but with five minutes left Williams crossed for his second of the day, and Aironi's fourth, but Olivier again missed the conversion.
And in the final move of the game Czekaj picked up his second try for the bonus point, Sweeney duly adding the extras with his seventh successful kick of the day to seal the win.
|Aironi Rugby Score Card|
|Cardiff Blues Score Card|