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.