The Spaniards lead Group F with 20 points from eight matches, while Sweden are second on 15 points. Romania sit third with 14 points, followed by Norway with 11, reports Efe news.
Spain came into Tuesday's contest at a packed Friends Arena in Stockholm after experiencing their first stumble of the qualifying campaign: a 1-1 draw with Norway in Oslo.
The visitors set the pace from the start and Swedish goalkeeper Robin Olsen had a busy first half, turning aside a shot by Fabian and denying both Rodri and Gerard Moreno from point blank range.
La Roja enjoyed 71 percent possession and Sweden's first chance didn't come until after the 20-minute mark, when Mikael Lustig forced a stop from David De Gea.
The Swedes began to have more of the ball and De Gea was tested in the 33rd minute by a Robin Quaison header.
Swedish defender Victor Lindelof made a diving block when Fabian threatened for Spain just before the break and the sides went to the dressing rooms deadlocked 0-0.
The home side took the lead five minutes after the re-start courtesy of Marcus Berg's first international goal in more than a year.
De Gea stopped Berg's header as well as Emil Forsberg's effort on the rebound, but the ball ended up back with Berg and the keeper was helpless to stop it from going in the net.
The Spain keeper made another notable stop against Forsberg before leaving the match injured with 30 minutes left.
That Forsberg effort proved to be Sweden's last threat of the night, yet Spain struggled to convert their dominance into an equaliser.
La Roja still trailed at the end of regulation time, raising the possibility that they would be held scoreless for the first time in 38 matches.
Two minutes into stoppage time, Rodrigo nudged Fabian's low blast into the net to avoid the loss and guarantee Spain's presence in next summer's tournament.
Sweden can clinch the second spot in Group F with a win over Romania next month in Bucharest.
( With inputs from IANS )