The Liverpool target netted from the penalty spot with 10 minutes remaining to leave Tim Sherwwod's side with a tricky Europa League last-32 return leg.
Jan Vertonghen was ruled to have fouled Matheus with 10 minutes left on the clock, and Liverpool's January target Konoplyanka beat Brad Friedel in the Spurs net to leave Tim Sherwood's side with an uphill battle in the return leg at White Hart Lane.
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 the Premier League side with work to do but it could have been a different story had Roberto Soldado not spurned a golden chance when he missed an open goal 10 minutes into the second half.
By the end, 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-chance of the game in the 13th minute, with 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 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 4 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 defense.
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 equalizer 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.