The 24-year-old watched Hanover's 1-0 win over Nuremberg on Friday night in the Bundesliga and will train with his new team on Monday.
Having spent two seasons at Old Trafford and made three appearances for Senegal, Diouf failed to break into United's first team under Sir Alex Ferguson and spent last season on loan at Blackburn Rovers.
"I am glad that we now have another alternative for our attack with Mame Diouf," said Hanover coach Mirko Slomka.
"We are now going to have to integrate him quickly so that he can help us with our upcoming challenges in the Bundesliga and the Europa League."
Diouf, who joined United from Norwegian side Molde FK in 2009, will compete with ex-Germany striker Jan Schlaudraff and Norway's Mohammed Abdellaoue for a place in the Hanover attack.