Dutch champions Ajax returned to the Amsterdam Arena on Thursday trailing 1-0 from last week's last 16 first leg tie.
But Frank de Boer's men levelled the aggregate score on the hour mark when Riechedly Bazoer capitalised on slack defending from Egidio.
Dnipro remained competitive despite being shorn of suspended trio Roman Zozulya, Ruslan Rotan and Artem Fedetskiy and head coach Myron Markevech remaining in Ukraine with illness.
And they wrested the initiative when Konoplyanka curled home from the edge of the box in the seventh minute of extra time to stun the home crowd.
A nervy ending ensued when Van der Hoorn nodded in the rebound when the returning Kolbeinn Sigthorsson's initial effort rebounded off the crossbar in the 117th minute.
There would be no further drama, though, as Ajax's last realistic hope of silverware this season came to an end - with De Boer's side 11 points adrift of leaders PSV in the Eredivisie.
The hosts started brightly and Anwar El Ghazi provided Arkadiusz Milik a chance which was wasted before the teenage winger went close himself, forcing Denis Boyko into a fine fingertip save.
Dnipro started to show more going forward and Nikola Kalinic was almost rewarded for a great piece of ingenuity when his 25-yard volley on the turn brought out the best of Ajax goalkeeper Jasper Cillessen.
Ajax's threat remained and Ricardo Kishna jinked his way into the area and forced a near-post stop from Bokyo, before Milik flashed a header wide from Nicolai Boilesen's pin-point left-wing cross.
Having dominated much of the game, the hosts made the breakthrough on the hour.
Egidio was caught in possession deep in his own half and Bazoer raced clear before sliding his shot beyond Bokyo, who allowed the ball to squirm beneath him.
Dnipro saw a great chance to restore their aggregate lead go begging as Roman Bezus nodded wide from eight yards, while Kalinic stung the palms of Cillessen.
Extra time came as neither side could muster a threat, but Dnipro finally found their range seven minutes into the additional period as Konoplyanka picked up the ball in the left channel and drove at the defence before bending a right-foot shot into the far corner.
In need of two goals, Ajax poured forward and they ensured a frantic finish as Van der Hoorn followed up after Van der Hoorn was denied.
Boyko's over-zealous celebrations at the final whistle prompted an angry response from Van der Hoorn, as Ajax frustrations were laid bare.
