Ivanisevic broke Costa's serve in the second game of the first set, but Costa broke back, then broke Ivanisevic again in the 11th game for a 6-5 lead.
But Chelsea attacked with more purpose after the break, with Costa scoring from close range and Eden Hazard drilling low beyond Kasper Schmeichel.
As a series of running battles broke out, Costa was fortunate to stay on for a shove on Marquinhos, having been booked earlier for a wild lunge on Silva - the game degenerating into a niggly, fractious affair.
During a break in Costas' telephone segment with Sandusky, the accused's lawyer, Jopseh Amendola, appeared in studio and took several questions.
The U.S. team was clearly frustrated in trying to break down the Costa Rican defense but it prevailed in the end thanks to a goal that was the result of a beautiful sequence of four great individual plays.
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Costa broke Ivanisevic in the fifth and seventh games of the second set, finishing the match in 70 minutes.
Costa broke away on the last climb of a hilly trek to secure the second stage win of his Tour career, crossing the line 42 seconds ahead of Christophe Riblon.
Villa had a header saved and Gabi smashed a fierce effort wide before Costa broke the deadlock just after the hour.
