The budget clothing chain occupied a floor of the eight-storey building, and some of the workers injured and killed in the incident worked for a company that supplied the brand.
In a statement released on its website, a Primark spokesman said: "Primark's team in Bangladesh has been working to put in place immediate and long-term help for victims of this disaster.
"We have partnered with a local NGO to address the immediate needs of victims, including the provision of emergency food aid to families. This initiative began in Bangladesh immediately (when) the extent of the disaster became clear.
"Primark will also pay compensation to the victims of this disaster who worked for its supplier. This will include the provision of long-term aid for children who have lost parents, financial aid for those injured and payments to the families of the deceased."
The eight-storey Rana Plaza building in the capital Dhaka housed thousands of workers when it collapsed on Wednesday.
The owner of the premises has been arrested and is being questioned on charges of negligence, illegal construction and forcing people to work.
Primark said it would be "reviewing our commitments constantly" to ensure they meet the needs of the victims, and also urged other retailers who used suppliers based in the building to offer assistance.
Ruth Tanner, campaigns and policy director at the anti-poverty charity War on Want, said: "If UK high street chains like Primark had put in place proper measures to ensure the workers who make their clothes are safe, these deaths could have been avoided.
"While Primark has taken some responsibility, the retailer and the other companies involved must pay full compensation, including loss of earnings, sign the Bangladesh Fire and Safety Agreement and ensure such a disaster never happens again."
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