The United States midfielder, who went on as a substitute, sent the ball high over Claudio Bravo's goal to the joy of Betis' fans.
The final at a packed Estadio La Cartuja in Seville offered a pulsating contest dominated for periods by both sides.
Manuel Pellegrini's Betis got an 11th-minute opener from a header by Borja Iglesias, who led its attack all night. Sergio Canales and Juanmi Jiménez both hit the post, and Valencia needed goalkeeper Giorgi Mamardashvili to deny Nabil Fekir and Iglesias in the final moments of regular time.
Valencia relied on its counterattacks, which produced an equaliser in the 30th when Hugo Duro chipped the ball over Bravo on the break.
It stayed 1-1 at the end of regulation and through 30 minutes of extra time, when the scoring chances dried up as the players visibly wore down.
It was Betis' third Copa del Rey title, ending a 17-year wait for a major title.
"This is what we wanted, to win the Copa del Rey again," said Betis' 40-year-old captain Joaquín Sánchez, who received the trophy from Spain's King Felipe VI.
"This is thanks to the work not only of this year. We have grown little by little and are having a great season."
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