Ukrainians Shakhtar - the 2009 UEFA Cup winners - held a slender advantage going into the second leg having scored an away goal in the Czech Republic last week.
But goals from Daniel Kolar and Milan Petrzela - scored within a four-minute period and from Plzen's only shots on target - left Shakhtar chasing the game and Luiz Adriano's late effort was not enough to prevent a 3-2 aggregate loss.
After winning 2-1 on the night, Dusan Uhrin Jr's men will next face either Lyon or Chornomorets.
It was a cagey first half at the Donbass Arena until, somewhat bizarrely, the very first shots of the game yielded two goals in quick succession and gave Plzen the upper hand.
In the 29th minute, Stanislav Tecl was dispossessed in the Shakhtar box and Kolar was quick to pounce on the loose ball before slipping it past the shell-shocked Andriy Pyatov.
The Ukrainians' grip on the tie loosened further in the 33rd minute as Petrzela made it 2-0 to the visitors.
The Czech winger was given all the time and space he needed to run at the Shakhtar defence before picking his spot and beating Pyatov from outside the box.
Shakhtar now knew they needed three goals to stay alive in the competition and came close to reducing the deficit when Bernard picked out Taras Stepanenko from a corner.
The midfielder got his head to the ball from near the centre spot but visiting keeper Petr Bolek looked lively and snaffled his glancing effort.
Shakhtar threatened once more when play resumed, Alex Teixeira forcing Bolek into making a catch with his left-footed drive.
Plzen were more than content to sit back and defend their lead, though, and Shakhtar were feeding on scraps for much of the second half.
They finally went close to testing Bolek again in the 77th minute as Stepanenko shot wide from just inside the box.
That near miss spurred them on to greater things and with six minutes remaining Stepanenko wriggled into the area to sting Bolek's palms from close range.
Adriano had cut a frustrated figure in front of his own fans but gave them some reason to cheer in the 89th minute when converting a simple finish from Vyacheslav Shevchuk's cross from the left.
The consolation goal had arrived too late, however, and Shakhtar bowed out of the tournament.