The irony in the story "The Machine that Won the War" comes from Asimov's manipulation of the "Technology as Protector and Savior" myth. In a nutshell, that mythic belief states that no problem will ever be so insurmountable that a technological advance can't be found or made to fix the problem.
In Asimov's story, that's what the general population believes. They are at war with the Denebians (from the planet Deneb), and have been at war for a long time. Humans have invented a supercomputer called "Multivac" to help analyze enemy strategy, create tactics, etc. With the computer's vast processing power, the people of Earth now think they should be able to win the war.
The story begins just as the war has ended and three important men are talking to each other. Lamar Swift, Executive Director of the Solar Federation, has led the main war forces. John Henderson, chief programmer, was in charge of Multivac's input data. Max Jablonski, chief interpreter, was in charge of interpreting Multivac's output data.
The irony of the entire story is that none of them paid any attention to the machine at all. They simply used their own intuitions about the information they were getting and giving. Henderson admitted that the data that was coming in from the field was so biased by the people giving the reports that he was forced to guess what should go into the machine. Jablonski admitted that the machine wasn't even working properly by the end of the war, and the Multivac reports weren't credible or even decipherable. So he guessed what info should be passed on to Smith. Smith then admitted that he never trusted the machine in the first place, so when it came to important decisions, Smith flipped a coin.
So while the entire world is thinking that technology has been their protector and savior, it was really three men's intuition, personal decision making, and blind luck.