The 21-year-old has scored 19 goals in two seasons for the Italian side and his arrival may help to fill the hole in the Spurs side which will be left if Gareth Bale's potential world-record move to Real Madrid goes through.
Roma published a statement on their website which read: "Roma announce they have signed a contract with Tottenham for the outright sale of the playing rights of Erik Lamela, in exchange for 30m euros.
"The agreement also includes the obligation to pay 5m euros in bonuses related to the achievements of the player and the English club."
Lamela, who can play on the wing or as a forward, has been capped three times by Argentina and will join fellow new arrivals Roberto Soldado, Paulinho, Nacer Chadli and Etienne Capoue in a new-look Tottenham side this season.
Lamela admits he will leave Roma with more than a few regrets - but he is ready to succeed in the Premier League with Spurs.
The youngster believes he could have done more to help the capital club improve on consecutive seventh and sixth-placed finishes during his two seasons.
He told the Gazzetta dello Sport: "I want to thank and bid farewell to the fans who always treated me well. I embrace them all and I'll always keep them in my heart.
"I'm proud to have played for Roma and I hope to come back one day. Maybe I didn't manage to achieve everything I wanted to, or could have.
"It's also true that we were lacking something as a team but I hope it goes better this year. I can see that all the lads have the desire to well and they're very focused.
"I hope the club, coach and my team-mates can go as far as they possibly can.
"For me, I hope to do well in London with Tottenham - and I'm ready."
Roma have already moved to replace Lamela, having signed Serbian forward Adem Ljajic for 11m euros from Fiorentina.