The Tottenham star continued his scoring surge while captaining the United States for only the second time in his 93 caps with the team.
Dempsey found the back of the net in blizzard conditions after Jozy Altidore's attempt from 18 yards out deflected off Costa Rica defender Roy Miller. It was the Tottenham forward's team-leading seventh goal of the 2014 World Cup qualifying cycle.
"It seemed like (Dempsey) really liked the snow," Jurgen Klinsmann joked after the game.
Dempsey was named captain of the team for the first time under coach Jurgen Klinsmann after the coach decided to leave regular captain Carlos Bocanegra off the roster. Dempsey had never worn the armband for a full match despite 92 caps leading to Friday's game.
"He's proud to have the captain armband and he was obviously thrilled to score that goal right away and he's leading by example," Klinsmann said. "He's ready for this role, he's ready to step it up as a real leader for this team. For a player with 93 caps, it's about time he's become a captain too."
Dempsey is the fifth player under Klinsmann to be named captain. Bocanegra, Tim Howard, Jermaine Jones and Kyle Beckerman have also worn the armband during Klinsmann's 24 games as coach.
"It's an honor," Dempsey said. "It's my second time being captain but my first time being captain from the start of the game. To play a part in captaining the team in an important game and to get a good result feels good. I feel like I did my job, but all the credit goes to the boys. They fought hard tonight."