Shakhtar, playing their first competitve match since early December, ended stubborn resistance from a makeshift Spurs side in the final 12 minutes at a freezing Olympiyskiy Stadium.
A gaffe from Tottenham's goalkeeper Heurelho Gomes handed Shakhtar the lead after 78 minutes with substitute Yevgen Seleznov scoring just seconds after coming on.
Jadson swung a freekick into the box and fellow Brazilian Gomes was beaten to the ball by Seleznov who headed in.
Gomes, who appeared not to be touched by any player, fell to the ground clutching his head and was given treatment while a chorus of boos rang out from the home fans.
Shakhtar left Spurs with an uphill battle for next week's return at White Hart Lane by adding a second goal two minutes from time. Jadson cut in from the right and played a one-two before creating space for a shot that comfortably beat Gomes.
Tottenham face a Premier League clash at Hull City on Monday in a battle to avoid relegation and a League Cup final against Manchester United at Wembley three days after the return leg but defender Michael Dawson said they were still in the tie.
"The last 15 minutes have cost us... we had coped well until then and we are disappointed to go home with a two-goal deficit," Dawson told Chennel Five television.
"It's going to be an uphill task but we still think we can do it. We know how good we are at home and we will have to take the game to them."
NEAT MOVE
Swedish midfielder Kennedy Bakircioglu's goal on the hour secured Ajax Amsterdam, the 1992 UEFA Cup winners and four times European champions, a 1-0 win over Fiorentina at the Artemio Franchi stadium in Florence.
Bakircioglu finished off a neat move in the penalty area to leave Fiorentina, semi-finalists last season, a tough task in the second leg.
Fiorentina coach Cesare Prandelli told LA7 television: "We had chances but we were not on form in front of goal. I'm convinced that we can win (at Ajax)."
Alberto Gilardino had a great chance to equalise midway through the second half but the ball got stuck under his feet when he was free just eight metres out.
Twente Enschede, second in the Dutch championship, continued their good run of form under manager Steve McClaren with a surprise 1-0 win over Olympique Marseille at Stade Velodrome.
Austrian international Marko Arnautovic hit the only goal after 22 minutes, bundling in Blaise Nkufo's left-wing cross.
Substitute Martin Vingaard headed an injury-time equaliser to earn FC Copenhagen a 2-2 draw with Manchester City at a snowy Parken stadium in the Danish capital.
City had twice led with their first goal on the half-hour coming after Copenhagen keeper Jesper Christiansen allowed a tame shot from Nedum Onuoha to squirm under his hands.
Brazilian Ailton Almeida equalised with a powerful header from a corner but the Danes were
