Jack suffered tendon damage after falling during the 30-3 EDF Energy Cup semi-final defeat at the Millennium Stadium and required surgery.
The 29-year-old All Black will miss the Heineken Cup quarter-final against the same opponents on April 6 and the rest of the Guinness Premiership campaign.
Saracens director of rugby Alan Gaffney said: "It's a blow for the club and more importantly for Chris, who has quickly settled in England.
"He has been producing some very good performances for us, but it's something that we have to deal with.
"Even though he won't be able to get out on the pitch, with his knowledge and experience he still has a very important role to play for us."