Luis Enrique's side enjoyed long spells of possession - a staggering 85% - and had the best of the chances, in the Group E encounter in Seville, but could not find a winner.
Alvaro Morata was guilty of squandering Spain's best opportunity when he fired wide with just Robin Olsen to beat.
An obdurate Sweden side executed their tactical plan to stifle Spain superbly and did threaten at the other end with Alexander Isak hitting the post.
Spain initially looked to open the game from wide areas as they clocked up the passes with Sweden completing just two passes in the opposition half inside the opening 20 minutes.
A teasing cross from Koke brought Spain their first real chance as he picked out Dani Olmo only for a superb diving one-handed save from Olsen to keep out his close-range header.
Koke whistled a shot just past the post with an angled effort struck with the outside of his boot, but should have done better on the half hour when he lifted a shot over the bar from a good position.
A mistake by Marcus Danielson let Morata burst through on goal, but with Olsen rushing off his line the striker curled his shot wide while Olmo went close again from long range.
Sweden flickered into life moments before half-time when a rare foray forward saw Isak's shot bounce off the knee of Marcos Llorente and strike the post.
Spain's play lacked intensity in the second half and some trickery by the impressive Isak created a golden opportunity for Marcus Berg on the hour, but he fluffed his shot from close range.
Olmo saw a shot blocked by Danielson, but Spain were unable to rediscover their tempo and failed to really test Olsen as the game meandered to a stalemate.
Gerard Moreno could have won it with a header from fellow substitute Pablo Sarabia's clever cross inside the six-yard box, but Olsen saved instinctively with his legs.
Then at the death Sarabia failed to get a clean touch on a dangerous ball into the box with the goal at his mercy.