The biggest expense when traveling to Japan is airfare so I always book the flight first. If money is no object, you can skip this and book your flight the way you normally do. But if you want to save money, start by opening an incognito window in Google Chrome or private browsing in Safari (this doesn’t track your cookies) and check Google Flights.
Enter your home airport and for the destination airport, enter NRT, for Narita. Hit enter then click on the little bar-graph symbol. You will get a graph of the daily lowest fares. Scroll through to the month you are thinking of traveling. The darker blue bars are the weekend and shortest bar is the lowest fare. The day the lowest fare falls on could change if you come back in a week or so. In my experience, though, they typically fall on either a Monday, Wednesday or Sunday. Now look at the field labeled Lowest Fares for a __ day trip. Play around with this number an see how the bars go up and down. This will tell you when to fly for the best price.
Once you have selected the days you want to fly you can search for the best place to book your flight. Here are some search engines to try. You can book from Google Flights, but, I have found lower fares elsewhere. I have not found lower fares on different days.
The biggest price difference is not which search engine you use, it is what day you choose to fly. For example, I did a search using Google Flights that told me that if I fly on Tuesday July 15, it will cost me $1516 but if I leave the next day, it will only cost $894. Then, using Kayak, I found a fare of $791 on July 16. So the most dramatic savings is dictated by the dates you fly.
On another note, I told you to fly into Narita. That is the biggest, and therefore the cheapest, airport . It is not the closest to Tokyo so if you can afford it, Haneda is much more convenient.