Podolski signalled the move was going ahead on Saturday after publishing a picture of himself alongside Inter head coach Roberto Mancini, but an official announcement was not forthcoming.
The forward trained with Inter on Monday and his parent club confirmed later in the day that a deal had been struck between the two sides.
His move may have come too late for him to start Inter's league clash with Juventus on Tuesday, but Mancini did hint that the 29-year-old may feature on the bench at the Juventus Stadium.
Inter will hope he can provide a much-needed spark to their season after finishing the first half of the campaign in 11th position – 18 points behind leaders Juve.
And the striker – who was part of Germany's World Cup-winning squad in Brazil last year – will be keen to get regular football after a frustrating campaign at the Emirates this season, with just seven substitute appearances in the Premier League.
Since arriving in England, Podolski has scored 31 goals in 82 appearances.
