The world, European and Italian champions said in a statement he had signed until June 2015 and that forward Jonathan Biabiany has joined Samp as part of the deal.
Media reports said 12 million euros had also changed hands.
"It's a great surprise. I've arrived yet I don't really understand what's happened," the 26-year-old told Inter Channel.
"It's happened so quick and I absolutely did not expect it but I am really happy."
Pazzini's arrival follows Inter's struggles during the long spells when last term's club top scorer Milito, 31, has been injured this term.
Stand-ins Goran Pandev and Biabiany have failed to make the grade either in the three-pronged forward line favoured by former coach Rafa Benitez or Leonardo's two-man attack, leaving the side fifth in Serie A.
Pazzini fits much better into a two-man attack with his superb heading and goalpoaching ability and is a numerical replacement for the more temperamental Mario Balotelli, sold to Manchester City in the close-season.
His purchase is yet more proof of Inter buying Italian rather than their previous approach of almost always looking abroad.
Centre-back Andrea Ranocchia arrived from Genoa earlier in the January transfer window.
Pazzini, who has scored six leagues goals this term, is just the kind of player Benitez was crying out for at the start of the season.
The ex-coach's public demand that Inter bought in January, an outburst which ultimately cost him his job in December, has been granted to new boss Leonardo after his good start.
Mid-table Samp's decision to sell Pazzini, having also let their other top forward Antonio Cassano join Milan, has stunned their fans even though on-loan Manchester United striker Federico Macheda and ex-Palermo man Massimo Maccarone have already been brought in during Janauary.
Inter are also in the process of selling midfielder Sulley Muntari to Sunderland with Genoa's Houssine Kharja lined up as replacement, media reports have said.
