The 23-year-old Dane propelled from the left to hold off Nikias Arndt (Giatnt-Alpecin) and Jean-Pierre Drucker (BMC) over the line in Gandia.
It is the third victory for Orica-BikeExchange at this year's Vuelta after Simon Yates and Jens Keukeleire grabbed wins on stage 6 and 12 respectively.
“It’s really fantastic, a big dream come true,” Cort said. “Our main goal is the general classification and protecting Esteban Chaves and Simon Yates, but every now and again we get to give it a try.”
“We saw the same with Jens Keukeleire on stage 12 and today it was my turn to give it a go and it was really amazing. This is fantastic for me to win a stage in my first Grand Tour ever.
“This is a great team to be part of at the moment, three stage wins and two guys in the top five overall, it’s pretty special.”
The overall classification remains the same as the peloton heads into the stage 19, 39km time trial.
As it happened
After just six kilometres, Pierre Rolland (Cannondale-Drapac), Fumiyuki Beppu (Trek-Segafredo), Quentin Jauregui (ag2r-La Mondiale) and Mattia Cattaneo (Lampre-Merida) pulled clear of the peloton and were soon joined by Louis Vervaeke (Lotto-Soudal).
The quintet's advantage reached a maximum of seven minutes, 100 kilometres from the finish. With an eye on the sprint finish, Giant-Alpecin then worked hard to diminish the gap which dropped to two minutes with 35 kilometres to go. Two kilometres later, Cattaneo was dropped and the remaining escapees were caught 11 kilometres to the line.
Jan Bakelants (ag2r-LaMondiale) tried a doomed attack with 3.5 km remaining but he was quickly caught by the bunch.