Padovani went over for the match-defining try behind the posts with barely 60 seconds left but it stemmed from a remarkable run by fullback Ange Capuozzo, who darted through the Welsh defense and produced an outrageous sidestep before an inside pass to his winger.
They still needed Garbisi to convert and he did it before some emotional celebrations from him and the team.
"You'd think we'd won the World Cup," said Italy coach Kieran Crowley, who took charge last year and had beaten only Uruguay in his short tenure.
The Italians still finished in last place in the Six Nations for the seventh consecutive year, but ending with a first ever win in Cardiff — and their first win over Wales in 17 attempts — will make it feel like a successful campaign.
Their only two wins in 57 previous away games in the Six Nations were in Scotland, in 2003 and '07.
As for Wales, an unexpected title last year has been followed by a fifth-place finish and being on the end of one of the biggest shocks in the competition's long history.
It was a day when stalwart lock Alun Wyn Jones made his 150th appearance for Wales — an unprecedented achievement for a single country — and flyhalf Dan Biggar reached 100 Welsh caps.
There was nothing to celebrate at fulltime, though. Biggar, the captain, could barely summon up any words after the match to describe how he felt.
"It's just not good enough," he said.
It looked like Wales might escape from an error-strewn performance with a narrow win when Josh Adams scored a 69th-minute try, after bursting off his left wing and piercing the Italian defense, that was converted for 21-15.
Before that, the British and Irish Lions winger produced a massive, try-saving tackle in the left corner to stop Italy winger Montanna Ioane going over in the 45th minute, when the visitors were 12-7 ahead and proving more than a match for Wales.
But the Italians still had one big moment in them. In particular, the impish Capuozzo, in his first full test, who collected the ball inside his own half in front of a wall of red shirts.
He skipped to his right and sped through a gap between Taulupe Faletau and Adams — who had just been named man of the match — in a diagonal run toward the right touchline. Straightening up, he flummoxed covering defender Kieran Hardy with a sidestep that left Italy with a two-on-one. His inside pass to Padovani was perfect and the winger did the rest.
Garbisi and Padovani kicked the rest of Italy's points with penalties.
Center Owen Watkin scored Wales' first try in the 28th, with Biggar's conversion making it 7-6.
Most will have thought that was the platform for Wales to march to a customary win over Italy.
Not this time.
