The home side drew level in the 58th minute when Adam Nemec got his foot to a long pass ahead of an onrushing goalkeeper at the edge of the penalty area and the ball bounced back to Mak to slot into an empty net.
Three minutes later Mak latched onto a pass just outside the penalty area and his shot deflected off an Icelandic defender into the net to give the home side a 2-1 lead.
Michal Duris completed the scoring in the 84th minute when his left-footed shot beat the keeper after an Icelandic defender lost the ball deep in his own half.
The Slovaks will make their first trip to the European Championship finals as an independent nation after finishing second in Group C behind Spain. They booked their ticket in October after netting three goals in six minutes to beat Luxembourg.
Iceland qualified for the finals of a major tournament for the first time after a 0-0 draw with Kazakhstan in September, finishing as runners-up to the Czech Republic in Group A.
(Reporting by Michael Kahn,; Editing by Steve Tongue)
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