Newcastle United striker Mitrovic will be looking to add to the 12 goals he scored on loan at Fulham last season, while Matic will be key in midfield alongside Lazio's Sergej Milinkovic-Savic.
The two other teams in Group E, Brazil and Switzerland, meet in Rostov later Sunday.
The former Chelsea defender made his 104th global appearance, surpassing the previous record of 103 by former midfielder Dejan Stankovic.
They fashioned half a chance when Tadic played a pass to Mitrovic, who produced a sublime touch to bring it down but saw his shot blocked.
Costa Rica started positively with Bryan Ruiz slipping Marcos Urena through on goal in the second minute, but he failed to beat the keeper in a one-on-one.
The final moments of the match were marked by a squabble on the sidelines as Nemanja Matic got into a tussle with a Costa Rican coach.
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It's a daunting enough task playing or managing at the World Cup with eyes from around the globe on you, but Serbia manager Mladen Krstajić would have had extra reason for being nervous.
He was put through with four minutes left but stumbled over the ball and Navas gathered comfortably. Navas also stopped Milinkovic-Savic's bicycle kick in front of goal in the 42nd minute, but it was ruled offside.
Costa Rica play another short corner, but Guzman gets the cross in this time and Gonzalez gets his back-post header all wrong.
The scoreline hardly reflected Serbia's dominance against 2014's surprise quarter-finalists, though it was Costa Rica who went closest in the first half.
Now with Italian club Lazio, rumours have swirled that Manchester United might be interested in the dynamic 23-year-old midfielder.
Vucic told the team: 'I don't think it's such big money for you... but we wanted to show you how important it is to our country that you should represent it the best way possible'.