Probably yes. You did not tell us how many dependents your cousin has. If you look at the I-864 form, you will see that USCIS considers not only the people he has sponsored but also the size of his household. The following URL takes you to the I-864P form which shows the amount of income he must have. http://www.uscis.gov/files/form/i-864p.pdf
There are other things besides income that can be used (house equity, etc). Talk to an experienced immigration lawyer.
You should always consult with an experienced immigration attorney to make certain that the advice you received is appropriate for your particular immigration case.
If by "sponsor" you mean file an affidavit of support, then Yes if he meets the income requirements.
J Charles Ferrari Eng & Nishimura 213.622.2255 The statement above is general in nature and does not constitute legal advice, as not all the facts are known. You should retain an attorney to review all the facts specific to your case in order to receive advise specific to your case. The statement above does not create an attorney/client relationship. Answers on Avvo can only be general ones, as specific answers would require knowledge of all the facts. As such, they may or may not apply to the question.
Look at I-864P and when you counting the household members at to you and him and all the minors who live with you the five family members. You will know your answer.
NYC EXPERIENCED IMMIGRATION ATTORNEYS www.myattorneyusa.com; email: email@example.com; Phone: (866) 456-8654; Fax: 212-964-0440; Cell: 212-202-0325. The information contained in this answer is provided for informational purposes only, and should not be construed as legal advice on any subject matter.