+1(514) 937-9445 or Toll-free (Canada & US) +1 (888) 947-9445
http://www.cic.gc.ca/english/immigrate/sponsor/relatives-apply-who.asp

Who can be sponsored

You can sponsor:

* parents
* grandparents
* brothers or sisters, nephews or nieces, granddaughters or grandsons who are orphaned, under 18 years of age and not married or in a common-law relationship
* another relative of any age or relationship if none of the above relatives could be sponsored, and you have no other relatives who are Canadian citizens, persons registered as Indians under the Indian Act or permanent residents and
* accompanying relatives of the above (for example, spouse, partner and dependent children).
 
No, you can't.

The only category your wife's cousin could *possibly* call under is:
* another relative of any age or relationship if none of the above relatives could be sponsored, and you have no other relatives who are Canadian citizens, persons registered as Indians under the Indian Act or permanent residents.

But, as you mentioned... you have a wife who is most likely a Canadian Citizen or Permanent Resident. (Sorry to assume, I could be wrong on that of course, since you didn't clarify your wife's status).
 
Even if you go by "any other relative" clause, your wife would have to do it because it's her cousin. She could not have any other relatives in Canada or any other relatives that can be sponsored, that is, if she has parents who are alive, even if they have no interest in coming to Canada, she could not sponsor her cousin.

If you end up in Saskatchewan or Manitoba, she could sponsor the cousin under PNP if you and the cousin meet the requirements of that provinces PNP. Alberta has a family PNP too but they don't allow cousins.

Manitoba program here: http://www2.immigratemanitoba.com/browse/howtoimmigrate/pnp/pnp-family_support.html
Saskatchewan program here: http://www.immigration.gov.sk.ca/family/?Anc=7e8f4b1d-79b4-4120-8b3a-599d97fc7875&Pa=ebd9edd0-fecd-4503-b6f7-70a7366de4c4