From Longman Dictionary of Contemporary Englishbe on the housebe on the houseFREE/COST NOTHINGif drinks or meals are on the house, you do not have to pay for them because they are provided free by the owner of the bar, restaurant etc → house
Examples from the Corpus
be on the house• On New Year's Eve, he offered a glass of champagne to everyone, on the house.• The owner let us have the first bottle of wine on the house.• First drink is on the house. 