In the Importance of Being Earnest really the battle of the sexes is only reflected towards Act III when the personas of Jack and Algernon are exposed, the women declare themselves 'scorned' and plan a plot to get back at them for lying. In this last showdown, the men actually ignored the women and instead stayed outside in the garden eating biscuits and muffins rather than follow the angry women inside the house for arguing.
Right after they ate and returned back in, Wilde made a synched dialogue when the women would say the same things to the men at the same time, and then vice versa. Certainly at the end of the showdown, the men came out victorious particularly when it is obvious to the reader that Gwendolyn and Cecily are utter idiots.
Wilde did all this on purpose, to extend his already professed ideas of what marriage was. He detested the notion of it and everything so much that he basically mocked it all over the play. For this reason, it is expected that the women will come out losing, and the men -which in the end even got their wish to be Ernest- came out uber victorious.