Burak found the net with a powerful header in the fifth minute, running on to a free kick from Hakan Calhanoglu, who last week scored Turkey's first ever goal against England.
Turkey, whose 2-1 loss to England at Wembley followed a 14-game unbeaten streak, face holders Spain, Croatia and the Czech Republic in Group D at the tournament which starts on June 10.
Slovenia, who hosted Sunday's game in Ljubljana, failed to qualify for the finals after losing a playoff to Ukraine.
(Writing by Daren Butler; Editing by Ken Ferris)
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