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Paraguay looked set to claim all three Group B points in Rio de Janeiro after taking a fourth-minute lead thanks to Oscar Cardozo's penalty, with Derlis Gonzalez scoring an unstoppable second.
But invited guest Qatar, which won its maiden Asian Cup earlier this year, showed character to fight back and star man Almoez Ali got it back in the game with a curler from outside the box.
When Khoukhi's shot squeezed through Paraguay goalkeeper Roberto Fernandez, the backtracking Rojas could only nod the ball in before getting tangled in the net and seemingly damaging his shoulder.
Qatar made a chaotic start in its Copa debut, almost conceding in the first minute before Pedro Miguel gave away a penalty for blocking a Bruno Valdez header with his arm.
Paraguay striker Cardozo had not played at the Copa since 2007 but he stepped up to smash the resulting spot-kick straight down the middle to open the scoring.
Qatar should have levelled in the 16th minute but, arriving unmarked at the back post, Hassan Al Haydos somehow failed to convert a low left-wing cross from Abdelkarim Hassan.
Almoez Ali was denied by Roberto Fernandez on the stroke of half-time, with Paraguay seeing a second goal for Cardozo disallowed by VAR within five minutes of the restart.
Paraguay would not be denied in the 56th minute, however, as half-time substitute Gonzalez unleashed a 25-yard rocket that curled away from the desperate dive of goalkeeper Saad Al Sheeb.
Scoring nine goals at the Asian Cup marked Ali out as a man to watch and he unleashed a terrific hit from 20 yards to make it 2-1 and spark hope of a comeback.
When Rojas lacked composure in front of his own goal with 13 minutes to play, Qatar had earned a priceless Copa point to justify its inclusion.
