After a slow start from both teams, Quaison gave the Swedes the 33rd minute lead, stabbing Ludwig Augustinsson's pass home from close range to notch his third goal in his seventh senior international.
The 25-year-old Stockholm native, who grew up a few stops on the metro line from the national stadium, was also involved in Sweden's second seven minutes later.
He fired a dangerous ball into the box that was cleared as far as winger Viktor Claesson, who rifled the ball home from the edge of the area.
Romania upped their game in the second half and pulled a goal back through substitute Claudiu Keseru in the 58th minute as he stooped to head home Razvan Marin's chipped pass.
The Romanians pressed forward but created little in the way of goal-scoring chances, and the Swedes looked dangerous on the break, with Claesson shooting into the side netting after a late counter-attack.
The arrival of promising young Swedish forward Alexander Isak in the dying minutes was greeted with rapturous applause by the crowd of 30,115, and he wasted no time getting off a shot on goal which was steered behind for a corner.
Sweden travel to neighbours Norway for their next qualifier, while Romania host the Faroe Islands.
