For Quins, this result means that they could still technically be dragged into the relegation fight, having gone into the contest only nine points above the drop zone.
Leeds opened the scoring through Van Straaten with a penalty in the first minute, but Quins hit back with a penalty from Paul Burke to level the scores soon after.
Van Straaten then hit another one to snatch the lead back, and the home side extended that lead midway through the first half when wing Chris Hall went over for the game's opening try.
Van Straaten converted, but a penalty from Burke on the stroke of half-time brought the visitors slightly closer.
Two minutes into the second half they were rewarded for their endeavour when England wing Dan Luger – fresh from scoring England's final try in their Grand Slam win in Dublin less than a week ago – crossed the line for a try for the Londoners.
Burke added the extras, as well as two further penalties and a drop-goal, while a third penalty for Van Straaten got them back in the mix.
As the clock ticked down, it looked for all the world that Quins would walk away with the points, that was however until Albanese crossed the line right at the death for the vital try, Van Straaten holding his nerve to stick away the crucial conversion.
For Leeds Tykes:Try: Hall, AlbaneseCons: Van Straaten 2Pens: Van Straaten 3
For Harlequins:Try: LugerCon: BurkePens: Burke 4DG: Burke