Stockton's success gave Utah a dramatic 103-100 victory in sixth game of the Western Conference finals.
It capped Utah's comeback from a seven-point deficit in the final two minutes and put the Jazz into the final against the Chicago Bulls. The first game is on Sunday in Chicago.
This was Utah's fourth trip to the conference finals in six years, and the Jazz had earned a reputation as a team that could not win the big game.
Stockton put an end to that by taking over the game in the final quarter. He finished with 25 points and 13 assists in his best game of the series, with 15 of his points coming in the final 12 minutes.
Houston took a 98-91 lead with 2:02 left but Bryon Russell answered with his third three-pointer of the night to pull Utah within four. Stockon then brought it level at 98-98 with 1:03 left.
Charles Barkley made two more free throws with 32 second remaining. Stockton tied the scores once more with 22.4sec left.
That allowed Houston to set up for the last shot but they missed. Jazz called a time out and Stockton calmly won the game.
It will be the first trip to the championship round for every member of the Jazz, from future Hall of Famers Stockton and Malone, with a combined 25 years of NBA experience, to veterans Jeff Hornacek (11 years), Antoine Carr (13 years) and the rest of Utah's players.
Even Jerry Sloan, the longest-serving coach in the NBA after nine years with the Jazz, will be going to the final round for the first time. He never made it in 11 years as a player or in three seasons as coach of the Chicago Bulls from 1979-82.