Konoplyanka, a target of Liverpool in the January transfer window, slotted home an 81st-minute spot-kick after Jan Vertonghen was adjudged to have brought down Matheus inside the box, condemning Tottenham to their first European defeat of the season.
It was sweet revenge for Dnipro coach Juande Ramos, who delivered Tottenham's last trophy by winning the League Cup in 2008, only to be sacked the following season with the club bottom of the Premier League table.
The result leaves Tottenham with work to do in the second leg next week and it could have been an altogether different story for Tim Sherwood's men, who saw Roberto Soldado spurn a golden chance when he missed an open goal 10 minutes into the second period.
By the final whistle, though, Sherwood was thankful to depart with his side trailing by only the solitary goal as Roman Zozulya was denied by the frame of the goal in stoppage time.
On a dry and rutted surface, the visitors created the first half-opening of the game in the 13th minute after a minute's silence was observed in respect to those who lost their lives this week during the recent political unrest in Ukraine, Soldado seeing his near-post header blocked by Dnipro goalkeeper Denis Boyko.
Dnipro ought to have gone ahead 12 minutes later when Matheus was played clean through on goal, but Brad Friedel spread himself to block the Brazilian's effort low down to his left, before thwarting the striker again two minutes later when he tried his luck from 25 yards.
Soldado wasted a glorious chance to break the deadlock in the 56th minute when, with the goal at his mercy, the Spaniard somehow cleared the crossbar from four yards after being picked out by Paulinho's ball across the box.
Friedel had to be alert three minutes later to parry Matheus' rasping effort from the edge of the box as the game became more stretched.
