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The Jaguares kept the 2012 and 2013 champions scoreless in the final 30 minutes of the game to set up a home semi-final against either the ACT Brumbies or Sharks.
Flanker Pablo Matera and winger Matias Moroni scored tries for the home side, who had to fight back from a 16-8 deficit early in the second half when the Chiefs had looked to have seized the momentum.
Loose forward Lachlan Boshier scored the only try for the Chiefs, who had made an unlikely run to the playoffs after losing their first four games of the season.
The Chiefs, one of only two sides to beat the Jaguares at the Estadio Jose Amalfitani all season, were under pressure from the outset and a handling error cost them an early try, Matera touching down in the first minute.
The hosts dominated the early going and flyhalf Joaquin Diaz Bonilla's 21st-minute penalty gave them an 8-0 lead but the Chiefs got back into it after a darting run by scrumhalf Brad Weber led to Boshier's try.
Flyhalf Jack Debreczeni gave the Chiefs a 10-8 lead heading into halftime, which he extended with a second penalty after the break as the visitors' scrum started to get the upper hand.
Another scrum penalty, converted by Marty McKenzie after Debreczeni limped off with a thigh injury, gave the Chiefs a 16-8 lead but that only sparked the home side into action.
Stretching the Chiefs defence and challenged them at the breakdown, the Jaguares scored 13 unanswered points in eight minutes with winger Moroni crossing and Diaz Bonilla adding two more penalties for a 21-16 lead.
The Chiefs responded in the final 20 minutes but the Jaguares defence held firm, thwarting wave after wave of attacks and a vital turnover with less than five minutes remaining allowed scrumhalf Tomas Cubelli to clear from their goal-line.
With the Chiefs' lineout malfunctioning the entire game, the home side were able to regain control of the ball and hold on for the victory.