Costa Rica started brightly, with Vladimir Stojkovic diving to save at the feet of Marcos Urena, before Giancarlo Gonzalez headed straight at the Serbia goalkeeper. Kolarov's freekick is a contender for goal of the tournament as his effort, from further out than Cristiano Ronaldo's versus Spain, curled over the wall and into the top corner beyond Navas' reach.
Defeat for Costa Rica ended their run of five World Cup games without loss, having exited Brazil 2014 on penalties in the quarterfinal stage after drawing with Holland.
Ricardo Gareca's men threatened again two minutes later as Francisco Calvo cut in from the left and fired a low shot wide of the near post from 20 yards out, in what was the final action of the half.
The Roma defender reiterated that Costa Rica has a quick playing style and strong defenders, as well as well-known forwards.
The Lazio playmaker was Serbia's most creative influence, although his blushes were spared by the linesman's flag when he fluffed a first-half chance after racing through on goal.
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"It is great to have three points in the bag, but we are not finished. I don't want to do the maths yet", he said.
There was a late scuffle as Nemanja Matic tried to take the ball off a member of the Costa Rican staff on the sidelines, before a VAR red-card review saw Aleksandar Prijovic booked for a separate incident, but Serbia held on with relative ease for a potentially priceless win.
The former Chelsea player was tied with ex-midfielder Dejan Stankovic on 103 caps prior to the game, although Stankovic earned his caps under the various names of Yugoslavia, Serbia and Montengro and finally Serbia, as the country became independent. Navas also stopped Milinkovic-Savic's bicycle kick in front of goal in the 42nd minute, but it was ruled offside.
Serbia, under the tutelage of Mladen Krstajic for the first time in a competitive match, controlled much of Sunday's clash at Samara Arena, but could not find a way through Costa Rica's stubborn rearguard until Kolarov stepped just before the hour.
Now with Italian club Lazio, rumors have swirled that Manchester United might be interested in the dynamic 23-year-old Milinkovic-Savic..