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The first half-chance of the match came inside the opening five minutes when Vyacheslav Shevchuk delivered a dangerous cross from the left, but Seleznyov could not make contact inside a crowded Slovenia box.
The deadlock was broken in the 23rd minute when Yarmolenko twisted inside the Slovenia box before finding the bottom corner during a period where Garmash and Serhiy Rybalka also came close for the dominant hosts.
A fairly unsavoury first period, which saw Ukraine register three attempts on target to Slovenia's zero, also brought four yellow cards, with both teams guilty of play-acting during the opening 45 minutes.
Ukraine had their second of the match in the 54th minute, however, when Artem Fedetskiy delivered a low cross from the right and Selezynov could not miss as he turned the ball over the line from close range.
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Garmash had a wonderful chance to make it 3-0 just before the hour when he met a dangerous cross from the right, but his header was straight at Samir Handanovic from close range.
Seleznyov came close to his second of the match when he burst into the box in the 69th minute, but the attacker's effort hit the post, before the danger was cleared as the visitors continued to wobble in defensive areas.
Yevhen Khacheridi fired one over the crossbar in the 72nd minute as the pressure continued from Ukraine, with Slovenia struggling to make their mark despite a couple of substitutions. One of those subs, Nejc Pecnik, had an effort 15 minutes from time, but he cleared the crossbar from inside the box.
Slovenia had their moments late on, but the opportunities continued to fall to Ukraine, with substitute Ruslan Malinovskiy and Konoplyanka coming close to a third, but the hosts were ultimately made to settle for a 2-0 win, which makes them big favourites ahead of the return fixture, which will take place on Tuesday.