Duro's finish in the second half at San Mames leaves the tie firmly in the balance, with Valencia playing the second leg at home at Mestalla on March 2.
Athletic's Raul Garcia had given them the lead shortly before half-time but the 23-time winners of the Spanish Cup were unable to hold on for the victory.
The winner of the tie will play either Rayo Vallecano or Real Betis in the final in April. Betis defeated Rayo 2-1 at Vallecas in the first leg on Wednesday night.
Athletic knocked out Real Madrid in the quarter-finals and to win the cup this season would be particularly special, after they lost two finals in two weeks last year.
The first defeat was by Basque rivals Real Sociedad in a postponed final from 2020 before they lost again to Barcelona in the 2021 competition a fortnight later.
Athletic are the most successful Copa del Rey club left in the tournament given only Barcelona can better their 23 cup victories.
But of the four semi-finalists, Valencia have won it most recently after they beat Barca in 2019, under Marcelino, who is now in charge of Athletic.
The home side took the lead in the 37th minute when Garcia held off former Barcelona midfielder Ilaix Moriba at the back post to head in Iker Muniain's bending cross.
Valencia's substitute Maxi Gomez was sent off at half-time as a feisty 45 minutes ended with a confrontation between players from both teams. Gomez had yet to play in the game but will now be unavailable for the second
There was little between the sides after the restart and Valencia grabbed an equaliser just after the hour-mark as Athletic goalkeeper Julen Agirrezabala did well to parry Bryan Gil's finish but the ball fell straight to Duro, who bundled in.
Valencia thought they should have had a penalty in injury-time when Duro went down under a challenge from Athletic's Dani Vivian but referee Jose Munuera was not convinced.