House Bill 2284 was previously passed by the House, meaning it will now go to Gov. Jan Brewer for consideration.
The Senate approved the bill 17-13 along party lines.
Litigation is likely if the bill becomes law because requiring warrants for such searches is part of a 2010 lawsuit settlement.
Majority Republicans said the change is needed to ensure abortion clinics can be inspected without delays.
Sen. Nancy Barto, R-Phoenix, said abortion clinics are the only health-care institutions in the state that are not subject to unannounced inspections.
Democrats countered that removing the warrant requirement would open clinics to harassment by unfriendly regulators and could subject the state to even more lawsuits over its abortion rules. Democrats also say the bill is unnecessary, since only one warrant has been issued in the past four years.
