LAX has a lot of competition, but the best fares are likely to be mainland Chinese carriers with connections at PVG, PEK, CAN, etc. (TWOV applies.)

It might be cheap to fly to some country other than Japan, such as mainland China or Thailand, and take a LCC flight on a separate ticket to Tokyo (or elsewhere in Japan) but be sure you understand the risk and allow lots and lots of time between flights on different tickets.