The win gave Norrkoping 66 points, four more than IFK Gothenburg who could manage only a 2-2 draw at home to Kalmar, with AIK coming third on 61 points after a 1-1 draw with Orebro.
Elfsborg beat Falkenberg 4-2 to make the most of Malmo's misfortune, leap-frogging them into fourth place to snatch the final European place.
Going into the final day with a one-point lead over Gothenburg, Norrkoping -- who finished 12th last term and were rank outsiders when the season began -- knew a win would secure the title no matter what happened in the other games.
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Their title hopes received an immediate boost when Malmo captain Markus Rosenberg was sent off after five minutes for stamping on Linus Wahlqvist.
With the tension threatening to get the better of Norrkoping, Kujovic broke the deadlock on the half-hour mark, coolly slotting home the ball to put his side in the driving seat.
The goal did little to settle his side however. Malmo, who are currently contesting the group stages of the Champions League, lost Franz Brorson to a second yellow card but continued to press forward in search of the goal that would guarantee them European football next season.
Instead Kujovic struck again, splitting the Malmo defence in injury time with a pass to Arnor Traustason, who scored to send the jam-packed away fans into ecstasy as they celebrated the club's first league title in 26 years.
At the other end of the table Halmstad and Atvidaberg go down, while Falkenberg, who finished third from bottom on 25 points, will meet second-tier side Sirius in a playoff to decide which of them will play in the top flight next season.
(Editing by Clare Fallon)
