Spurs announced the deal on their official Twitter account this afternoon.
The club said: "We are delighted to announce that we've reached agreement with AS Roma for the transfer of Erik Lamela.
"Erik made a total of 67 appearances in all competitions for Roma, scoring 21 goals. He also has three full caps for Argentina."
Roma yesterday announced a deal was close on their official website, saying: "Roma announce they have signed a contract with Tottenham for the outright sale of the playing rights of Erik Lamela, in exchange for 30million euros.