The two nations will meet at Nouveau Stade de Bordeaux to kick off a group that also holds England and Russia.
Chris Coleman's side enjoyed a good qualifying campaign, finishing runners-up to Belgium and conceding just four goals in 10 matches, losing just once.
Their build-up to the tournament in France hasn't gone so well, as the Dragons failed to win any of their four warm-up games, having drawn 1-1 with Northern Ireland and losing to the Netherlands (3-2), Ukraine (1-0) and Sweden (3-0).
The Falcons also impressed during the qualifiers, beating group winners Spain 2-1 in the home leg to finish in the runners-up spot after suffering only two losses.
Jan Kozak's men will be a lot happier going into the Euros as they beat Switzerland (3-2), Iceland (3-1) and Germany (3-1), while drawing with Latvia (0-0), Republic of Ireland (2-2) and Northern Ireland (0-0) in the lead-up to the France showpiece.
The two teams have only met twice before, during Euro 2008 qualifying when the two encounters delivered 13 goals, with Slovakia winning 5-1 in Cardiff before the Welsh ran out 5-2 victors in Trnava.
In team news, Coleman has been boosted by the return to fitness of Joe Ledley, who has recovered from a broken leg.
The Crystal Palace midfielder sustained the injury on May 7 but has made a remarkable recovery and could see some action against Slovakia.
Coleman also welcomes back Liverpool midfielder Joe Allen from a knee problem and Reading striker Hal Robson-Kanu from an Achilles tendon injury.
Meanwhile, Kozak must choose between Adam Nemec or Michal Duris to lead the line after including just those two strikers in his 23-man squad.
Robert Vittek, his country's all-time top scorer with 23 goals, was omitted from the final party due to injury.