Molins struck in the 36th and 70th minutes to hand Malmo a victory that puts them on 60 points with one game left, giving them an unassailable five-point lead over second-placed AIK Stockholm. IFK Gothenburg are in third spot on 51.
Malmo's triumph books them a place in the second qualifying round of next season's Champions League, with AIK guaranteed a spot in the Europa League qualifiers.
The Allsvenskan's final Europa League qualifying spot will go to Gothenburg or Helsingborg, who are fourth on 49 points.
With Monday's match under threat from a storm warning issued by the Swedish weather service, Malmo travelled to Boras, 55 km east of Gothernburg, knowing a win would clinch the title.
They started brightly, peppering the Elfsborg goal with shots as Molins and Jiloan Hamad went close to breaking the deadlock in the pouring rain and gusting wind.
Molins finally put the visitors ahead before the break when he cut in from the left before smashing the ball into the top left corner of the net in front of the ecstatic Malmo fans.
Molins, who only returned to the club in August after an injury-hit two years playing in Belgium and Spain, doubled the advantage with 20 minutes left by calmly chipping the ball home from a tight angle to wrap up the points and the title.
(Reporting by Philip O'Connor in Stockholm; Editing by Ken Ferris)
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