Esat had rejected Telenor's bid, even though the two companies are partners in the Irish firm's mobile telephone offshoot Esat Digifone, where Telenor has a 49.5% stake.
Esat's board accepted the offer within 15 minutes of it being made. BT had given the firm three hours to ponder the deal.
BT acted as what is known as a "White Knight", rescuing the company from the unwanted attentions of another rival.
Co-operation with Norwegian rival
Esat had rejected Telenor's bid, even though the two companies are partners in the Irish firm's mobile telephone offshoot Esat Digifone, where Telenor has a 49.5% stake.
BT said that it was prepared to work with Telenor to ensure the success of this venture.
The UK telecoms firm said that its offer represented a premium of 17.6% on the bid made by Telenor at the end of last year.
Telenor has not yet decided whether it will make a counter bid or withdraw its offer.
BT wants Esat's chairman and chief executive Denis O'Brien and the existing management team to continue leading the business.
Ireland's number two
British Telecom described the acquisition of Esat as "an exciting opportunity to invest in a leading competitor in Ireland".
Mr O'Brien described BT as "a major player and a friendly buyer" and predicted that Esat could gain up to 50% of the Irish telecommunications market within three years.
Esat is the country's second largest fixed line and mobile phone operator.
The company has a strong internet presence. It is the number one player in the Irish corporate internet market and one of the leading providers of internet access to Irish consumers.
BT it already knows that its offer will be accepted by at least a quarter of share holders, through an agreement with Esat founder and chairman Denis O'Brien, and with US company Fidelity Investments.
In London, BT's share price fell 2.7% or 34 pence to 1179 pence.