Jones struck in the 49th minute and Dempsey added the insurance goal in the 80th minute as the USA advanced to face either Panama or El Salvador, who meet later Sunday to complete the semi-final line-up in the regional championship.
German-born midfielder Jones, making his ninth appearance for the USA, fired a shot from outside the area that was deflected by Jamaican defender Jermaine Taylor's outstretched leg past goalkeeper Donovan Ricketts. The United States gained a man advantage after Taylor was sent off in the 67th minute for a foul on Jones.
The United States made the most of the advantage as Dempsey scored to take the Americans lead up to two. A cross from Juan Agudelo found Dempsey alone on the left side of the goal. Ricketts came out to contest, and Dempsey eluded him and slotted the ball home. Agudelo had come on in the 12th minute for Jozy Altidore, who departed with a strained left hamstring.
Jamaica had reached the quarter-finals as the winners of Group B with victories over Grenada and Guatemala and Honduras. However, the Reggae Boyz had never beaten the United States and had never made it beyond the Gold Cup quarter-finals, trends they couldn't reverse on Sunday.
The semi-finals will be held on Wednesday in Houston, where Mexico and Honduras are already slated to meet. Defending champions Mexico advanced with a 2-1 victory over Guatemala on Saturday, when Honduras defeated Costa Rica 4-2 on penalties after the two teams were level 1-1 after extra time at New Meadowlands Stadium. The Gold Cup champions earn a berth in the 2013 FIFA Confederations Cup.