Brommapojkarna stood for a long time for a good effort at home and early in the second half the bottom team also cheered for a lead goal through Nikola Vasic.
But the goal was disallowed because Vasic had run down an Elfsborg player on his way to goal. BP coach Olof Mellberg was angry, but had to put up with a disallowed goal.
Instead, Emmanuel Boateng put the visitors in the lead through his first Allsvenskan goal when he neatly shot the ball in.
Before the game was over, the talented Jalal Abdullai had made it 2-0 and Simon Hedlund added another after a fine individual performance.
As Malmö lost points against Kalmar, Elfsborg goes up in the league lead. BP is in trouble in the qualifying place with three points down to Degerfors who are second to last in the table.
Four rounds remain in Allsvenskan.