Drmic missed an open goal from two metres but made amends in the 84th minute when he ran on to Blerim Dzemaili's pass, cut inside his marker and fired past goalkeeper Andre Blake.
The Swiss struggled to create chances against physical, well-organised opponents with a performance which does not auger well for their hiopes at next month's World Cup where they must face equally rugged Honduras in much more difficult conditions.
They were also frustrated by Blake who made several outstanding saves.
Switzerland were largely restricted to long-range shots in the first half including two from Admir Mehmedi which whizzed past the woodwork from 25 metres.
Haris Seferovic was also close from long range while Blake reacted brilliantly after Jamaica defender Adrian Mariappa deflected Stephan Lichtsteiner's cross towards goal.
Jamaica, with nine of their starting lineup based in England or the United States, occasionally looked dangerous as they broke forward.
The Swiss should have gone ahead in the 65th minute when Reto Ziegler's low cross found Drmic unmarked at the far post but he somehow scooped the ball over the unguarded net.
Blake then made another reaction save to deny Drmic from close range and dived at Lichtsteiner's feet to block the rebound, before finally being beaten with six minutes left.
The Swiss, who also face France and Ecuador in their World Cup group, host Peru in another friendly on Tuesday.
(Reporting by Brian Homewood; Editing by Ken Ferris)
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