Available convenient options to Savannakhet are:
1 (expensive). Air trip to Vientiane (ThaiSmile, 3500baht, about 18:30PM) followed by air trip to Savannakhet early in the morning on the next day (Lao Airlines, 1500-2000baht, 7:50AM) and taxi to the Thai embassy. You will be in front of the Thai consulate right before its opening. Before entering the consulate you should cross the road and check-in at the Macchiato de coffee hotel for 800 baht/night. This is because they will need your passport to copy on check-in.
2 (cheap). Air trip to Nakhon Phanom (800-1200baht) followed by bus trip to Mukdahan and catching a hourly bus to the Savannakhet bus station which is situated near the Thai consulate. Before entering the consulate you should cross the road and check-in at the Macchiato de coffee hotel for 800 baht/night. This is because they will need your passport to copy on check-in.
Another convenient options - air trip to Ubon or Roi Et (1000 baht) followed by bus to Mukdahan and bus to Savannakhet.
I have not mentioned any options which include long bus trips because in my opinion they are far from being convenient.