Captain Sergio Ramos scored for the seventh time in eight internationals to give his side the lead with a penalty in the 64th minute and Alvaro Morata also converted from the spot to double the home side's advantage in the 85th.
Substitute Oyarzabal completed the victory at Real Madrid's Santiago Bernabeu stadium two minutes later after coming off the bench to make his competitive debut, curling the ball into the bottom corner from just inside the area.
The win took Spain to 12 points at the top of Group F after four games. Sweden are second, level on seven points with third-placed Romania, who thrashed Malta 4-0 away.
Spain thought they had taken the lead in the first half when striker Rodrigo converted a cross by Valencia team mate Dani Parejo from close range but the goal was wrongly ruled out for offside as video technology is not used in the qualifiers.
Spain coach Luis Enrique was absent for the third game in a row for personal reasons but they always looked in control without truly sparkling until Oyarzabal's moment of brilliance.
TWO PENALTIES
The hosts continued to wear the dogged Swedes down and they got their breakthrough when visiting defender Sebastian Larsson's outstretched arm blocked the ball and Ramos calmly deceived keeper Robin Olsen to convert from the spot.
Sweden barely managed to trouble Spain goalkeeper Kepa Arrizabalaga, who kept his place again at the expense of David de Gea, and the visitors conceded another penalty when substitute Morata was tripped by Filip Helander.
Ramos passed up the opportunity to score again and offered the ball to Morata, who also made no mistake from the spot, although Real Sociedad forward Oyarzabal got the biggest cheer of the evening for his impressive strike.
"We struggled a bit at first but in the second half we tried to break them down more and we've taken another step towards the Euros," said goalscorer Ramos, who dedicated the win and Friday's 4-1 victory over the Faroe Islands to Luis Enrique.
"This is an abnormal situation but at least we've managed to give a win to our coach. It was very important to get these six points and hopefully after the holidays everything will return to normal and we can all restart this journey again together."
Romania took charge against Malta with an early double from George Puscas and a 34th minute goal from Alexandru Chipciu, who was sent off late in the game before Dennis Man completed the rout in stoppage time to leave the Maltese on three points.
Fourth-placed Norway have five points following a 2-0 win away to the Faroe Islands who are bottom with no points.
(Reporting by Richard Martin, editing by Pritha Sarkar and Ken Ferris)
