The Germany international forward has left the Gunners and join the Turkish giants on a three-year contract.
According to BBC Sport, Galatasaray had to pay Arsenal an initial transfer fee of £1.9 million for the services of the 30-year-old.
Wales international midfielder Ramsey has now taken to Twitter to wish Podolski the best of luck in Turkey.
Podolski joined Arsenal from German club Koln in the summer of 2012 and scored 31 goals in 82 appearances for the Gunners.
The former Bayern Munich forward struggled for playing time at the Emirates Stadium last season and was sent out on loan to Italian outfit Inter Milan in the January transfer window.
Galatasaray won the Super Lig title last season and will feature in the group stage of the Champions League in the 2015-16 campaign.