There were slender Coppa Italia victories for Fiorentina, Genoa, Parma and Udinese, but Verona needed penalties to beat Venezia after a 3-3 thriller.
Fiorentina welcomed the first goals for Lorenzo Venuti and Jose Callejon, but Santini performed a stunning overhead kick to get Serie C side Padova back into it at 2-1.
Goalkeeper Emanuele Terracciano had to make a last-gasp save on Nicastro’s header to prevent it going to extra time.
Fiorentina will face Udinese in the Fourth Round, as the Friulani beat Vicenza 3-1 with goals from Fernando Forestieri, Gerard Deulofeu and Ignacio Pussetto.
Genoa set up a fascinating Derby della Lanterna with Sampdoria in the next round, but needed a Gianluca Scamacca brace to get the better of Serie C side Catanzaro.
Parma thanked a Yann Karamoh brace and Andrea Adorante’s late finish to beat Pescara 3-1.
They will face Cosenza in the Fourth Round next month.
Verona were pushed all the way, as they were held 2-2 by Venezia, then it ended 3-3 in extra time, until goalkeeper Pandur took centre stage with two saves in the shoot-out.
SPAL also required penalties to get the better of Serie A outfit Crotone, as it was 0-0 after 90 minutes and 1-1 after 120, so they will face Monza in the next stage.
Crotone 1-1 SPAL (4-5 on pens)
Cigarini pen 107 (C), Seck 113 (S)
Genoa 2-1 Catanzaro
Scamacca 7, 65 (G), Fazio 83 (C)
Hellas Verona 3-3 Venezia (6-4 on pens)
Ilic 41 (HV), Salcedo 59 (HV), Johnsen 84 (V), Capello 87 (V), Modolo 98 (V), Vieira 110 (HV)
Fiorentina 2-1 Padova
Venuti 10 (F), Callejon 32 (F), Santini 55 (P)
Parma 3-1 Pescara
Karamoh 26, 43 (Pa), Adorante 70 (Pa), Nzita 91 (Pe)
Udinese 3-1 Vicenza
Forestieri 21 (U), Deulofeu 60 (U), Pussetto 64 (U), Gori 88 (V)