The club did not disclose the financial terms of the deal, but the midfielder is reportedly worth 7 million pounds ($11 million).
Sigurdsson was set to join Swansea, where he spent a successful stint on loan during the second half of last season, but the deal fell apart after the club's manager, Brendan Rodgers, left to join Liverpool in May.
After Rodgers' move, the attacking midfielder was said to have wooed by both Tottenham and Liverpool, with Spurs sealing the deal just a day after Villas-Boas joined the London team.
“We are delighted to announce that we have completed the transfer of Gylfi Sigurdsson from Hoffenheim,” Tottenham said in a statement. It did not say how long Sigurdsson will play for the club.
Sigurdsson scored seven goals in 19 appearances for Swansea, gaining a reputation for his well-timed runs from midfield and his lethal shots from outside the box.
Whether he's joining at White Hart Lane to replace Croatian playmaker Luka Modric remains to be seen.
Modric, who plays in a similar position to Sigurdsson, has been linked with a move away from Tottenham this summer after the club failed to qualify for next season's Champions League.