Sarries beat Tigers in the final round of pool matches, and will play Aaron Mauger's side at Allianz Park again.
Quins, seventh in the top-flight, go to an Exeter side hoping to reach the final for the third time in a row.
Welsh regions Dragons, Scarlets, Ospreys and Cardiff all failed to make it into the last four.
The Anglo-Welsh Cup did not take place last year due to the 2015 Rugby World Cup.
The semi-finals will take place between 10-12 March, with the final to be staged at the Twickenham Stoop.
Reigning champions Saracens will host Leicester, with Exeter at home to Harlequins in an all-Premiership semi-final line-up in the Anglo-Welsh Cup.