Vucinic scored the winner in the 67th minute, chipping the ball home from a narrow angle. He was then booked for taking his shorts off and putting them on his head.
The win gave Montenegro, whose national team was formed only three years ago, a maximum nine points from three games after beating Wales and Bulgaria.
Switzerland, who have been on a downhill run since beating eventual winners Spain at the World Cup, have lost their two matches so far.
A Swiss television commentator immediately compared the result to their home defeat by Luxemburg in a World Cup qualifier two years ago.
Montenegro, who leaped into the top 50 of FIFA's world rankings last month, dominated the first half in a battle of two teams who are expected to fight for the runners-up spot behind England in the group.
The Swiss, however, had a flurry of chances just before the break as the hosts' goalkeeper Mladen Bozovic saved shots from Alexander Frei and Gokhan Inler in quick succession.
AS Roma forward Vucinic put the hosts ahead before Bozovic was called into action again, stopping a low, powerful free-kick by substitute Eren Derdiyok and then denying Frei, Switzerland's record international scorer.
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