La Roja — whose unbeaten streak in World Cup qualifiers stretched back 66 games — took the lead inside four minutes when Jordi Alba lifted a brilliant ball into the box for debutant Carlos Soler to strike into the net.
But the lead barely lasted a minute as Isak won the ball outside Spain’s box shortly after play had restarted and fired a shot into the bottom corner.
Sweden’s second came just over 10 minutes into the second half as Spain failed to clear from a corner; Dejan Kulusevski was able to find Claesson, who hit a shot past Unai Simon’s near post.
Spain had an opportunity to equalize through Adama Traore, while Isak was also denied the chance to score his second goal by a superb tackle from Aymeric Laporte. However, the final score remained 2-1 in Sweden’s favor.
Sweden is now on a record run of seven straight home wins in World Cup qualifiers, which also includes victories over France and Italy.
It moves to the top of Group B with Spain second ahead of Kosovo, Greece and Georgia.
Sweden’s victory means England, which beat Hungary 4-0 on Thursday, now has the best current unbeaten streak in World Cup qualifiers, stretching back to 2009.