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.
Andros Townsend made his first start for Tottenham since mid-December after recovering from a hamstring injury, while 16-year-old Josh Onomah took his place on the bench for the first 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.
Ten minutes before the break, Ivan Strinic was booked for hauling down Paulinho on the edge of the area, with Tottenham arguing that he should have been sent off for denying a goalscoring opportunity. Townsend sent the resulting free-kick a yard wide.
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.
Sherwood introduced Christian Eriksen off the bench in place of Townsend in the 64th minute and the Denmark international almost made a telling contribution with his first touch, but his slotted finish beyond Boyko was ruled out for offside.
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.
Soldado saw a volley blocked before Friedel was involved again, rushing from his line to deny Giuliano after the Brazilian had jinked his way through the Tottenham defence.
The deadlock was eventually broken when Vertonghen brought down Matheus and Konoplyanka coolly tucked home the penalty in off the left-hand post to give Dnipro a crucial first-leg advantage.
Tottenham pushed hard for an equaliser as Paulinho forced a ricochet that struck the base of the post before substitute Harry Kane failed to get a telling connection with the goal at his mercy.
It could have been so much worse for Tottenham, though, as Zozulya sent a header against the post in stoppage time.
