Also included in the proposal is an aparthotel, exhibition centre, gallery and a medical centre.
The 1.32-hectare site at Grand Canal Harbour in Dublin 8 consists of disused industrial and commercial buildings and a warehouse, which is a listed structure.
The company plans to demolish all the buildings except the warehouse and construct six blocks ranging from three to 12 storeys.
Dublin City Council's granting of permission is likely to be followed by an appeal to An Bord Pleanala, given the level of objection to the scheme.
Dozens of opponents came out against the ambitious plan.
Heritage body An Taisce, while saying the site was suitable for a "high density scheme", insisted the plans were "excessively crowded and constrained".
It added: "The proposal conveys a sense of developing the former canal harbour (to) within an inch of its life."
Diageo Ireland expressed its support "for the principle of a mixed use redevelopment of these lands".
However, the company, the Guinness parent group, questioned the impact the towers might have on views from the company's rooftop Gravity Bar, one of the city's top tourist attractions.
In addition, a number of residents associations lodged objections.
Residents of Westcourt on James's Avenue complained locals did not know how "construction traffic, pollution, noise and construction hours will be managed".