Head coach Frankie McAvoy said Hearts "cannot dwell" on their early European exit as they were hammered by a rampant PAOK in a one-sided play-off second leg in Greece.
Any Hearts hopes of progressing to the Conference League group stage for a second successive season were extinguished early on after two quick goals from Taison and Brandon Thomas extended the Greeks' aggregate lead.
Giannis Konstantelias walked in a third before Taison strolled in for his second to emphasise the gulf between the two sides.
"The goals we conceded were pretty poor," McAvoy conceded. "It's difficult for me to stand here try and defend that because we didn't do well enough.
"It's a sore night for us. We need to put it to bed. It's gone now. That's down to failings from last season, not finishing third. We can't dwell on it.
"We've fallen short. We've got to concentrate on ourselves. We need to make Tynecastle a hard place for anyone to come, and try and pick up our form away from home - which hasn't been good enough, if I'm honest."
Hearts rarely troubled the PAOK goal throughout the night and must now watch on as Aberdeen, Celtic and Rangers are left to fly the Scottish flag in Europe.
Even the most optimistic of Hearts fans would have struggled to anticipate their side being involved in the Conference League group draw on Friday.
And what optimism still existed before this second leg was swept away when Taison's low free-kick went through the legs of Lawrence Shankland and snuck into the far post - although there appeared to be a strong shout for offside that went unanswered by the officials.
Alas, it would not have made much impact on the final outcome. Seven minutes later, Thomas hooked the ball over his shoulder for a magical second.
The third goal was painful for the travelling Hearts support. A simple long ball was left by Kye Rowles, allowing Andija Zivkovic to burst down the right and find an unmarked Konstantelias.
The 20-year-old was allowed two bites of the cherry as he trickled a shot past Zander Clark, with Alex Cochrane only able to watch on as the ball bounced into the net.
PAOK's final punch saw them catch Hearts on the counter. Taison and Thomas traded passes before the former cantered into the box to roll past Clark.
Player of the match - Taison (PAOK)
Hearts' frailties give cause for concern - analysis
As is often the case, it was more the manner of defeat than the defeat itself that will worry Hearts fans. Unable to lay a glove on their opponents in an attacking sense, they looked vulnerable every time the Greeks came forward.
Part of that is understandable. They were roared on by a cacophonous, rampant home support in a deafening Toumba Stadium. They struggled with the conditions and the size of the pitch - larger than the Nou Camp, we are told.
But three - arguably all four - goals were preventable. Defenders too slow to close down, waiting for offside flags that never came.
One could defend them with a lack of experience - but plenty of these players featured in last season's Conference League group stage.
Of course, the concern would be minimal had they brushed Dundee aside at the weekend. Instead, they struggled to create and were picked off with a wonderful solo strike. They need to show real mental resilience now to drag themselves out of this rut.
What they said
Hearts head coach Frankie McAvoy: "I'm not here to make excuses. We changed it second half to try and be more direct and get the next goal in the game. If you get that you never know.
"The goals we conceded were pretty poor. The disappointing part for us was that we didn't keep the ball well enough, particularly first half. The first goal, I don't know what you say. The boy is offside. It was an uphill battle from there. The second goal is poor - we don't defend at the back stick or in the middle of the goal."
Hearts are back to Scottish Premiership action on Sunday with a visit from Motherwell (15:00 BST).