The Argentina international has joined the club for a reported fee of £30 million – surpassing the £26m paid for Valencia striker Roberto Soldado and, before that, the £17m spent on Brazil midfielder Paulinho.
Lamela, 21, scored 15 goals in Serie A last season and is widely seen as a replacement for Gareth Bale, who is expected to soon complete his world record move to Real Madrid.
The forward, who can play out wide or through the centre, is Tottenham's second signing of the day having followed Romanian defender Vlad Chiriches - who cost £8.5m from Steaua Bucharest - to the north London club.
The deal for Lamela takes Tottenham's summer spending past the £100m barrier.
It is believed the deal was processed in time for Lamela to make his debut against Arsenal on Sunday.