American Airlines has round-trip flights from San Antonio or Austin (SAT, AUS) to Tokyo (HND/NRT) for $919 with a connection. Flights August to February. Departures on Monday, Tuesday, Wednesday, Thursday, and Sunday. No Fridays or Saturdays. Extended weekends available.
Haneda is closer to Tokyo than Narita.
The 6-month moving average for the lowest fare found is about $1241. This fare is less than that, but most of the prices for Asia airfare this year has been higher than previous years.
A Carry-on bag and also a personal item is allowed for free on American. You can avoid paying for seat selection by accepting the randomly assigned seat at the gate.
Click for more details and booking links Cheap Flights: San Antonio or Austin to Tokyo, Japan (HND/NRT) $919 r/t [August-February] – American Airlines