The minnows are hosting the US in the final round on Wednesday and would have known they only had to match or better Mexico's result in Costa Rica.
But Jimenez's heroics appear to have rescued his country from a disastrous qualifying campaign and set them on a collision course with the All Whites.
Mexico can still sneak into third if they beat Costa Rica and Honduras lose to last-placed Jamaica, while Panama can claim fourth if they beat the United States and Mexico lose to Costa Rica.
If the current standings remain after Wednesday Mexico, the 21st-ranked nation in the world and with 14 previous World Cup appearances, will start as heavy favourites when the All Whites travel to the 105,000-seat Azteca on November 15. The return leg is set for Wellington on November 20.
The Hex
Standings
USA (19 points), Costa Rica (15), Honduras (14), Mexico (11), Panama (8), Jamaica (4)
The games
Wed: Costa Rica v Mexico, Jamaica v Honduras, Panama v USA