The 2022 World Cup hosts are one of two guest teams at the South American championship along with Japan.
However, their debut got off to a nightmare start when Paraguay's Oscar Cardozo smashed in from the penalty spot in the fourth minute.
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Qatar caused their opponents plenty of problems but after Cardozo had another strike ruled out after a VAR-review, the Asian Cup champions conceded again when Derlis Gonzalez scored early in the second half with a long-range rocket.
Striker Almoez Ali made amends for spurning an earlier sitter by pulling his side back into the game with a distance strike which looped into the net in the 68th minute.
Nine minutes later, Boualem Khoukhi surprisingly levelled with a scrappy finish to salvage a point for the underdogs.
Qatar face Colombia in their second match on Wednesday 19 June, before their final Group B match on Sunday 23 against Lionel Messi's Argentina.
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