Power, the pole-sitter, took the lead after the final restart of the 60-lap road course race, beating Briscoe to the finish line by 1.2 seconds to claim his series-leading third victory of the year.
Briscoe ended in second after passing defending champ Dario Franchitti on the final lap.
Franchitti finished third while Raphael Matos came in fourth and Mario Moraes completed the Top 5.
Power's triumph allowed him to expand his lead over Franchitti to 32 points. He also moved 40 points ahead of Franchitti's Target Chip Ganassi Racing teammate Scott Dixon, who ended up in eighth place at Watkins Glen.