The 21-year-old becomes Spurs' seventh signing of the summer, and second of the day after the arrival of Steaua Bucharest defender Vlad Chiriches, with the deal of around £30m breaking their transfer record for a third time.
Lamela started his career with River Plate in his homeland before joining Roma in 2011 for a fee of around £11m.
He scored four goals in his opening season in Serie A but made a massive breakthrough in 2012-13, when he hit the target 15 times in 33 games, second at the club behind recent Southampton signing Daniel Osvaldo.
Lamela has also won three international caps after progressing from their Under-20 set-up.
His fee surpasses the £26m Spurs spent on Valencia striker Roberto Soldado earlier this month, a deal that broke the mark they had set earlier in the summer when signing Brazil midfielder Paulinho.