Marta Cox converted a penalty in the 73rd minute, finishing both teams with four points in Group A but giving Panama the place in the knockout round based on goal difference.
Jamaica, which needed a victory to claim the spot, appeared headed for the final four with Shantel Bailey's goal in the 33rd minute. But a foul by Courtney Douglas on Karla Riley gave the 14-year-old Cox her opportunity to become a hero.
Panama's quarterfinal opponent will be decided when Mexico plays the United States on Monday in the last game for both in Group B. The semifinal winners and the winner of the third-place game will qualify for the U-17 Women's World Cup later this year.
Besides the teams playing their third match in five days, temperatures on the artificial surface at Estadio Cementos Progreso reached 136 degrees Fahrenheit (58 Celsius), according to local broadcaster Albavision.
Both sides had a relatively equal number of chances in the first half with Bailey putting a pair of shots side in the first six minutes and Cox driving a free kick from 25 meter just over the bar in the 12th.
Bailey, who was Jamaica's co-leading scorer in Caribbean qualifying with five goals, scored her first of the CONCACAF finals to put the Reggae Girlz ahead.
She had the ball at the top of the area and tried to play it wide to teammate Oshay Nelson-Lawes. But Panamanian defender Astrid Diaz was able to get a foot to it, and the ball took a fortuitous bounce across the 18, where Bailey volleyed a right-footed shot from 16 meters high inside the right post.
In the second half, Jamaica had the better of chances starting with Shantell Thompson's header wide off Nicole Broderick's cross in the 53rd and Thompson's free kick from 22 meters that Panamanian keeper Karen Chavarria had to leap to punch over the bar.
Nelson-Lawes had probably Jamaica's best chance minutes before Panama's equalizer when, unmarked, she volleyed a cross high over the bar.
But Yassiel Franco tested Jamaican keeper Chris-Ann Chambers with a right-footed drive from the arc in the 64th and Cox earned her third goal of the tournament after Douglas used a high foot to kick Thompson while challenging for a bounding ball.
Jamaica was beaten by Canada 6-0 in its opener on Wednesday, but routed host Guatemala 7-1 on Friday to finish with a goal difference of zero.
Jamaica won its opener 3-0 against Guatemala, but lost to Canada 4-0 and ended with a goal difference of minus-1 -- becoming the third and last Caribbean team to be eliminated.