Inter also benefited from a penalty which infuriated Udinese, while Antonio Di Natale scored for the hosts to join Roberto Baggio as Serie A's joint-sixth leading all-time scorer with 205 goals.
The first half ended goalless but Udinese were reduced to 10 men in the 40th minute when Maurizio Domizzi, already booked for dissent over a corner, was given a second yellow card for a late tackle on Hernanes.
Udinese's anger increased when Inter were awarded a penalty at the start of the second half for a foul on Mateo Kovacic by Danilo.
Mauro Icardi ignored the furious Udinese protests to put Inter ahead but the hosts levelled almost immediately when Di Natale, 37, burst between two players to fire past Samir Handanovic.
Udinese were down to nine before the hour when Emmanuel Badu was twice booked for dissent in quick succession.
Podolski came on in the 65th minute and within one minute scored his first goal since his move from Arsenal in January.
The German's first shot from the edge of the area was blocked but the rebound went straight to him and he took one touch before rifling an unstoppable effort into the top corner.
The win lifted Roberto Mancini's Inter to seventh with 48 points and revived their hopes of qualifying for the Europa League.
(Writing by Brian Homewood in Berne; editing by Toby Davis)
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