The deal was finalised after Torreira flew in to London following Uruguay's exit at the World Cup in Russian Federation, having helped his country reach the quarterfinals.
"He is a young player already with good experience, but who wants to keep on growing".
Arsenal arguably haven't had a decent defensive midfielder since the Invincible days of Gilberto Silva, but Emery and the new recruitment team around him look to have got themselves a top-class performer.
The player will not arrive until next month following a period of rest after the World Cup.
Arsenal have officially confirmed the transfer of Matteo Guendouzi from Lorient, their third new signing in the space of 24 hours.
Emery has made the tenacious tackler his fourth summer signing after bringing in Sokratis Papastathopoulos, Bernd Leno and Stephan Lichtsteiner.
In Uruguay, the word 'garra' has been associated with Torreira - who admits his lion tattoo on his upper right arm represents the "strength and spirit we Uruguayans have inside" - but it is his versatility that has alerted the attention of Unai Emery.
The 22-year-old midfielder will wear the No. 11 shirt at the club after passing a medical. I hope I can do great things here. Most importantly, however, Emery needs to simply steady the ship and try to avoid the limbo-type situation Manchester United endured after the departure of Alex Ferguson because, at this moment in time, that's the last thing Arsenal need.
The Uruguayan global, who impressed at this summer's World Cup in Russian Federation and featured in all of his country's matches, joins from the Serie A for a fee believed to be in the region of £25m. Now I am calm and I look forward to the future, which for sure will be very nice.
Ozil recently switched to number 10. The JOE World Cup Minipod semi-final preview hosted by Dion Fanning and Paddy McKenna is here.
England train ahead of World Cup semi-final
In 2002, 2006 and 2014 they were eliminated in the group stages, and they failed to qualify for the 2010 tournament. They've gone to two extra-times, and if you have to go to extra-time against Russian Federation .
NHC says Chris strengthens into a Category 2 hurricane
However by Thursday, it should begin to weaken and will likely become a strong post-tropical cyclone by early Friday. Although Hurricane Chris is headed towards Canada, it will still create severe weather for those on the U.S.