Lionel Messi gave Argentina -- who came into the match on a 36-game unbeaten streak -- an early lead from the penalty spot in Doha.
But Saleh Al-Shehri equalised early in the second half and Salem Al-Dawsari struck a sensational winner.
Argentina had three goals ruled out for offside in the opening period as their World Cup opener fell flat.
"It's clear we lost the game because of our own mistakes, above all in the second half," said Martinez, who plays for Inter Milan.
"In the first half we should have scored more than one goal, but that's the World Cup. Now we have to rest and think about what's ahead. Those are two finals now."
Two-time World Cup winners Argentina came into the tournament as one of the favourites but now face a battle to get out of Group C, with matches to come against Mexico and Poland.
"Without doubt (the next two games will be difficult)," said Martinez.
"Today was a difficult opponent too. We started like that and then in the second half we made mistakes. They make the difference in games. We have to correct them for the games to come."
"(It hurts) a lot. We wanted to start the World Cup with a win," he added.
Check the 2022 World Cup schedule and groups standings.
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