The Iranian team lost to Saudi Arabia's Al Ahli 2-1 at the King Abdullah Sports City, but won on the away goals rule after the aggregate score finished in a 2-2 draw.
Padovani scored the opener for Naft Tehran in the 32nd minute.
Omar Al Soma scored twice in the 36th and 43rd but it was not enough for the home side.
Korea Republic referee Kim Dong-jin added eight minutes of stoppage time to the game, but even that was not enough for Al Ahli to find a way back and Alireza Mansourian's Naft advanced.