The Czechs, who topped Group A in qualifying, scored first when defender Tomas Sivok headed in a corner 17 minutes into the first half.
The home side doubled the lead in the 63rd minute when Tomas Necid converted a penalty after Daniel Pudil was pulled down in the penalty area.
Serbia pulled a goal back in the 78th minute when Petar Skuletic beat Czech goalie Tomas Vaclik, before Ladsilav Krejci extended the lead to 3-1 minutes later when his shot dribbled between the keeper's legs.
Ondrej Zahustel rounded out the scoring in extra time when he headed in substitute Jaroslav Plasil's cross seconds before the final whistle.
The Czechs have been revitalised by manager Pavel Vrba, who took over in January 2014 following the team's failure to qualify for the World Cup finals in Brazil.
Midfielder Borek Dockal, who scored four goals in qualifying, also added some of the cutting edge the Czechs had been lacking in attack.
(Reporting by Michael Kahn, Editing by Greg Stutchbury)
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