The Czechs dominated possession but could not break down the Montenegro defence until the 63rd minute when Vaclav Pilar collected a pass at the edge of the penalty area from playmaker Rosicky and rifled a shot home.
The home side continued to press and Tomas Sivok grabbed the second goal in stoppage-time after another assist from Rosicky.
"We beat a very tough opponent, and we are of course very satisfied," Czech coach Michal Bilek told reporters.
"I am very pleased with the effort that the players gave and we have to repeat it [in the away leg] on Tuesday."
Arsenal's Rosicky gave Montenegro trouble all night and created a number of chances for the Czechs, who got a pre-game boost when goalkeeper Petr Cech decided to play with a specially designed mask after breaking his nose at the weekend.
Cech was forced to make a few saves at the start of the second half but for most of the game the Czechs kept Montenegro at bay and forced them into long-range efforts at goal.
Bilek made a few changes to the side, including leaving striker Milan Baros on the bench, but the strategy paid off.
The Czechs, who won the tournament in 1976 as Czechoslovakia, travel to Podgorica for the return leg.
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