Czech captain Tomas Rosicky set up Pilar for his debut goal in the national team in the 63rd minute and the 23-year-old sent a superb right-foot shot across the box and into the net past diving Montenegro keeper Mladen Bozovic. Besiktas sweeper Sivok added another with a stoppage-time header from Rosicky's free-kick that put the Czechs in a better position ahead of the second leg scheduled for 15 November in the Montenegro capital of Podgorica.
Montenegro, who held England to two draws in their qualifying group, created the first scoring chance of the game, but Juventus striker Marko Vucinic sent a volley from seven metres above the bar.
The Czechs dominated the first half in the chilly evening with the temperature just above zero centigrade. Arsenal midfielder Rosicky was the mastermind behind most of their efforts, but his teammates failed to capitalise on his passes in the first half. Ten minutes from halftime, Petr Jiracek blasted the ball headed by the Czechs' lone striker Tomas Pekhart above the bar from near the penalty spot in the home side's best chance.
The visitors put the hosts under pressure as the second half started, but a Simon Vukcevic shot from 20 metres was cleared by Czech keeper Petr Cech and Stevan Jovetic sent a volley wide. Cech was in the starting lineup despite doubts about his health after the Chelsea keeper broke his nose in a clash with teammate Ashley Cole in Chelsea's 1-0 win at Blackburn last Saturday. He played with a special carbon-fibre mask -- an addition to the headgear he has been wearing following a skull fracture five years ago.
Later in the second half, Jiracek narrowly missed the top left corner of the Montenegro goal as the Czechs started to gain the upper hand. Ten minutes from time, Pekhart failed to capitalise on a Bozovic blunder as the Montenegro keeper set him up for a header in front of an empty net.
The Czechs are close to making their fifth European championships in a row, while Montenegro, the youngest UEFA member that emerged in 2006 as the country gained independence from Serbia, are eyeing their first major tournament ever.