The hosts, managed by former Tottenham boss Juande Ramos, were awarded a spot-kick on 80 minutes when Jan Vertonghen was penalised for pulling back impressive striker Matheus just inside the 18-yard box.
Konoplyanka, who was chased by both Spurs and Liverpool on transfer deadline day, gave Brad Friedel no chance as his penalty went in off the upright to ensure his side will arrive at White Hart Lane for next Thursday's second leg with a slender advantage.
It could have been worse for Tim Sherwood's men but Roman Zozulya's stoppage-time header thumped the upright with Friedel beaten.
Roberto Soldado had earlier spurned the best chance of the game on 56 minutes when he somehow side-footed over the bar from close range after Paulinho's cross from the right found him free at the far post.
Dnipro supporters sang the Ukrainian national anthem during the minute's silence before kick-off, with black armbands also sported by both teams to honour those who have lost their lives during the political protests in capital Kiev in recent weeks.
Tottenham's first effort on goal came courtesy of a wayward volley from the boot of Paulinho, who was then guilty of losing possession and allowing the hosts to break.
Soldado had the best chance of the early exchanges as he glanced Nacer Chadli's cross towards goal, with Dnipro goalkeeper Denys Boyko equal to the header.
Friedel was almost caught out moments later as he rushed from his line to clear only to find Dnipro skipper Ruslan Rotan, who could not lob the American as he made his way back to goal.
Dnipro wasted the best opportunity of the first half as Matheus raced through behind the Tottenham backline and spurned the chance to put the hosts in front as he dallied before shooting straight at Friedel.
Rotan forced a better save from Friedel shortly afterwards as he arrowed in a low effort which the 42-year-old pushed straight into the path of Roman Zozulya, who lost his footing at the crucial moment.
Spurs also created a great chance as Danny Rose's cross fell to Chadli but the winger could not control his dipping drive.
Matheus skipped past a combination of Rose and Etienne Capoue before tumbling in the area, with referee Lahoz not buying the dive but opting against booking the Brazilian.
He did award a free-kick right on the edge of the Dnipro area when Matheus' compatriot Paulinho was felled by Ivan Strinic, with Townsend's dipping set-piece flying just wide.
Referee Lahoz was involved again on 55 as he deemed Zozulya had dived under pressure from Friedel following a fierce free-kick into the Tottenham box.
Soldado then missed the visitors' best chance when he spooned Paulinho's cross over from five yards out.
Sherwood introduced Christian Eriksen from the bench shortly after the hour and he thought he had made an immediate impact as he finished well only to see an offside flag raised on the far side.
Friedel kept the tie level as he did well to keep out Matheus' drive before Nabil Bentaleb became the next player to fall theatrically inside the box, with Lahoz again unmoved.
Ondrej Mazuch headed over a corner with 17 minutes to go before the game turned on a penalty decision as Lahoz was finally persuaded into pointing to the spot for the hosts.
Vertonghen brought down Matheus and Konoplyanka made no mistake with the penalty as his effort went in off the post.
Substitute Harry Kane could not turn home a couple of good chances before Friedel saved well from a deflected Konoplyanka free-kick.
Zozulya then hit the upright as Dnipro looked to further their lead, but they had to settle for just a one-goal advantage to take to England.