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.
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.