Skacel struck the opening goal of the game after 20 minutes to give Hearts the upper-hand, and substitute Pospisil settled it five minutes from time.
In a highly-charged encounter, Hearts showed first when Edgaras Jankuskas drilled a 25-yard shot narrowly wide.
Aberdeen had shooting chances themselves and Gary Dempsey was twice closed down as he shaped to crack the ball in.
Scott Severin had the first shot on target after 19 minutes, but Craig Gordon had no difficulty getting down low to his left to save.
Within a minute, Hearts had grabbed the lead. With Hearts fans appealing for handball against Russell Anderson, referee Mike McCurry waved play on.
And Skacel did not hesitate as he pounced on the loose ball to draw Ryan Esson before slotting the ball into the corner of the net.
The goal eased any concerns Hearts may have had, and they should have been further ahead by half-time.
Steven Pressley lashed the ball over from a Skacel corner after the Dons had failed to clear their lines.
Skacel had an even better chance nine minutes from the break after a Roman Bednar shot had been blocked. But Russell Anderson quickly closed him down and he fired over the top from 10 yards.
Aberdeen had a chance themselves at the end of the first half when Dempsey saw the goal opening up in front of him, but Andy Webster made a terrific block.
Paul Hartley had the best chance after some sloppy Aberdeen defending, but he shot wide from just 15 yards with Esson exposed.
The Dons lost skipper Anderson after 57 minutes when he had to go off with a facial injury which did not help their cause.
Jamie McAllister hit the post for Hearts late on with a volley before Pospisil put the result beyond doubt by firing a second into the roof of the net with five minutes left.