Yes. Non UAE nationals can Sponsor UAE Residence Visa. This is possible by what is called a ‘secondary sponsorship.’
A male resident who meets certain pre-requisite criteria of Salary and Accommodation may be able to sponsor residence visa for his parents, wife and children. This is also possible for a female resident provided she meets the criteria.
A domestic servant also could be sponsored by a resident visa holder provided he/she meets the residency rules.
Women can sponsor their children if she is widowed or divorced. Women can also sponsor their husbands. Few years back there were stringent conditions that a lady had to meet to sponsor her husband. Those rules have been made lenient.
The salary requirement for sponsoring your family keeps changing. Hence I do not want to put a ballpark figure on it. Whenever you come across this situation you are requested to contact Immigration call centre.
In order to sponsor your wife you require her valid Passport, Attested Marriage certificate and at least 6 photographs. The charges again vary; so please contact the Immigration department or any typing centre authorised to carry out preliminary typing. You can also sponsor your children below 21 years of age and unmarried daughters above 21 years.
As a secondary sponsor you are responsible for your dependents. This includes financial support, debts and all the other aspects. Sponsoring your family is your responsibility. Although many companies offer financial assistance while applying for residence visa and medical.