Elsewhere, Belgium thrashed Estonia, England thumped Hungary in a match marred by reported racist chanting and Germany claimed an underwhelming win against Liechtenstein in Hansi Flick's first match as coach.
Viktor Claesson's second-half strike, after excellent work from Juventus winger Dejan Kulusevski, put Sweden two points clear of Spain at the top of Group B with a game in hand.
Carlos Soler had given Spain a fifth-minute lead, but Alexander Isak levelled for the hosts just a minute later.
Only the group winners qualify automatically for the 2022 tournament in Qatar, with the runners-up facing a play-off.
Spain have reached every World Cup since last failing to qualify in 1974, but have plenty of work to do to continue that run.
"Our position is getting very complicated, we no longer have it in our own hands," admitted Spanish skipper Sergio Busquets.
"We were playing well, but we missed that bit of luck. We have to continue, there are still many matches."
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