The three points boosted the Americans into a first-place tie with Costa Rica and Mexico in the North American tournament that will determine three automatic qualifiers for next year's World Cup in Brazil.
Those three points appeared to be out of reach when Jamaica's Jermaine Beckford scored in the 89th minute to forge a 1-1 tie. The United States had led since the 30th minute when Jozy Altidore scored the opening goal.
With a tie appearing to be almost certain, the Americans won a corner kick that was directed to Michael Bradley.
Bradley spun away from a defender and found Evans with a pass that he converted into a shot that found the right corner of the net.
It was the second win for the United States in four qualifying games of what will be a 10-match tournament among the six regional finalists.
The Americans have seven points, as does Costa Rica after its 1-0 win over Honduras Friday and Mexico following its 0-0 road tie with Panama. Mexico, however, has played one more game than has either the United States or Costa Rica.
Panama has six points, Honduras has four and Panama is stuck on two. The top three teams earn a berth in the World Cup and the fourth-place finisher will meet New Zealand to determine the final qualifier.
Friday's game began a crucial stretch in the qualifying process. The United States will host Panama in Seattle next Tuesday and, one week later, will entertain Honduras in Sandy, Utah.