The breastfeeding stations would include breast pumps for expressing milk and fridges.
The bill also states that employers must give time for women to breastfeed.
Rachel Nyamai, the MP who proposed the bill, said it will help women work and at the same time improve children's health.
The law is not about dealing with a stigma around breastfeeding because breastfeeding in public areas is considered normal, says the BBC's Angela Ngendo in Nairobi.
Instead it's about making companies provide a conducive environment for working mothers, our correspondent adds.
Currently it is extremely rare for organisations in Kenya to have specific areas for women to breastfeed their children.
The same proposal was rejected by the previous parliament after business groups threatened to stop employing women if forced to provide breastfeeding facilities.
The Health Bill 2015 now needs the president's assent to become law.