Ukraine created a chance out of nothing in the last five minutes when a loose ball in the Spain box bounced favourably for Stepanenko, but his shot went just off target.
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Slovakia are now set to make their first trip to the Euros as an independent nation but had a few nervous moments after racing into a 3-0 lead.
He converted the winning penalty in a dramatic shoot-out against Italy in the quarter-finals and played from the start as Spain won the tournament with a 1-0 triumph over Germany in the final.
Russian Federation qualified in second place in Group G behind unbeaten Austria, with Sweden finishing third.
With Slovakia expected to win in Luxembourg, Ukraine knew they had to chase a victory themselves and the game became more open.
Roy Hodgson’s youthful England line-up wrapped up a 100 per cent record in their qualifying campaign with a straight-forward 3-0 win over Lithuania in Vilnius.
Hungary and Turkey are fighting for the final automatic spot as a third-place team.
Ukraine, who finished with 19 points, three behind Slovakia (22), will now feature in next month’s playoffs.
CESC FABREGAS missed a penalty for Spain as Chelsea boss Jose Mourinho watched from the stands.
Manchester United’s David de Gea says that he needs to perform at a consistent high level if he is to secure his place as Spain’s number one goalkeeper for Euro 2016.
Substitute Serhiy Rybalka tested De Gea late on, but Ukraine’s end product had let them down in the end and Gaspar’s 20th-minute opener proved to be the difference between the two.
Ross Barkley and Alex Oxlade-Chamberlain added to a Giedrius Arlauskis own goal to make it ten wins out ten for the Three Lions.
And Slovenia confirmed their place in the play-offs with an underwhelming 2-0 win over San Marino in Serravalle with second half goals from Bojan Perec and Nejc Pecnik.
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Qualification caps a dramatic turnaround under new coach Leonid Slutsky, who took charge of the team in August with the 2018 World Cup host struggling to qualify. It ended with 18 points, two less than Russian Federation and 10 behind Austria.
