The host nation, beaten by Group A rivals Colombia in the tournament opener on Friday, produced a devastating first half goal blitz to sink their rivals from the CONCACAF region on a warm night at Soldier Field.
Dempsey opened the scoring from the penalty spot before Jermaine Jones and Bobby Wood struck just before half-time to effectively settle the contest. Substitute Graham Zusi added a fourth three minutes from time.
United States Soccer Federation chief Sunil Gulati had fueled fresh speculation about Klinsmann's position only hours before kick-off, telling journalists: "No-one has iron-clad job security."
Klinsmann meanwhile showed he was in no mood to cede ground to his critics, picking the same starting line-up to face Costa Rica that had disappointed in the 2-0 defeat to Colombia. Klinsmann's faith was rewarded however with a superb first half display from his team.
Costa Rica's players protested but they could have no complaints about what was a clear-cut foul. Dempsey then ignored a scrum of Costa Rica players around him as he stepped up to coolly ram home the spot-kick, sending Patrick Pemberton the wrong way and becoming only the second US player to score 50 international goals.