Substitute Pilar was on hand to apply a simple finish one minute into time added on in the Group A opener at Generali Arena to punish a dreadful mistake from Daryl Janmaat.
Newcastle United defender Janmaat attempted to cushion a header back to goalkeeper Jasper Cillessen, but hit the post and midfielder Pilar could not believe his luck as he turned the ball in from just a couple of yards out.
Borek Dockal put the home side in front with a stunning strike on Tuesday and coach Pavel Vrba was able to mark his maiden competitive match in charge of the Czech Republic by securing his first win as an international coach courtesy of Pilar's last-gasp goal.
Hiddink bemoaned his side's "naive" defending in their 2-0 defeat against Italy last Thursday and they were at it again early on in Prague.
It needed an important intervention from De Vrij to block Vladimir Darida's shot after the visitors were caught napping when the Czechs took a short corner.
Hiddink made a tactical change seven minutes before half-time, with Luciano Narsingh replacing Veltman and Netherlands switched from three centre-halves to a back four.
The lively Depay came into the game brimming with confidence, having scored eight goals already this season for PSV, and he almost added an international goal to that tally when his long-range drive was tipped over by Petr Cech.
The Czech Republic were pegged back 10 minutes into the second half, though, as De Vrij rose highest to head home Daley Blind's inviting cross from the left.
Robin van Persie's eyes lit up when he was presented with a sight of goal, but would have been relieved to see the flag up for offside after Cech kept out his attempt on goal.
Pilar then had the final say as he converted from point-blank range after Janmaat got his header all wrong right at the death and was left red-faced.
