Former Netherlands midfielder Engelaar converted a cross from Mate Dugandzic after 61 minutes to settle a game low on quality.
Having been unable to win a match for the first three months of the campaign, rock-bottom Heart have been transformed since John van 't Schip took over from the sacked John Aloisi at the end of December.
Unbeaten since January 10, the club have now won four games in succession to sit just six points outside the top six, who qualify for the league finals, with seven rounds remaining.
The Roar, meanwhile, are now just four points ahead of second-placed Western Sydney Wanderers after suffering consecutive defeats for the first time in 2013-14.
After a bright start from the Heart, who spurned several presentable chances early on, there was little in the way of goalmouth action prior to the interval.
Engelaar broke the deadlock following excellent work from Dugandzic, who cut in off the right flank and nutmegged Luke Brattan before delivering a delicate pass to the far post.
Dugandzic could also have been awarded a penalty after appearing to have his legs taken away by Roar goalkeeper Michael Theo.
The Heart then missed opportunities to put the game to bed, before a last-ditch tackle from Patrick Kisnorbo prevented Besart Berisha from claiming a stoppage-time equaliser.
