Basel took the lead in the first leg of the quarter-final on 34 minutes through Delgado and he added a second goal four minutes later to put the Swiss side in command.
A last-minute goal from Valentin Stocker wrapped up the win to leave Valencia with a daunting task to qualify for the semi-finals ahead of next week's second leg at the Mestalla.
The match at St Jakob-Park was played behind closed doors as a result of the trouble among Basel fans which caused the second leg of their 2-1 aggregate victory over Salzburg in the last round to be suspended for 13 minutes.
As the players slowly became accustomed to the muted atmosphere, Marcelo Diaz gave Valencia a fright after 18 minutes with a shot on the turn from the edge of the area, which was fumbled by goalkeeper Vicente Guaita before he scrambled to regain control of the ball.
And Guaita came under fire again five minutes later from Fabian Schar's 25-yard daisy-cutter, which wormed past the keeper but also the left post.
However, Basel made their pressure count after 34 minutes with a superb goal from Delgado, who was teed up by Philipp Degen before lashing an unstoppable strike across the penalty area and into the bottom left corner.
And the Argentinian struck again four minutes later, making the most of Stocker's precision cross from the left wing to tuck the ball home from close range.
Valencia had looked ill at ease throughout the opening period but at least managed a decent shot before the interval, although Fede Cartabia's volley was a fraction too high.
Juan Antonio Pizzi's half-time talk appeared to hit home with the Spanish side, and Basel goalkeeper Yann Sommer made his first save of the match after 55 minutes, palming away a clean strike from Eduardo Vargas before Paco Alcacer headed the resulting corner over the bar.
Vargas had another good chance after 68 minutes with a stooping header which he directed just past the right post as Valencia tried to get an important away goal.
Sommer then pulled off a fantastic save with two minutes remaining, getting down low to block Seydou Keita's snap shot.
Stocker wrapped up the scoring in stoppage time, making the break from his own half and racing away before lifting the ball over Guaita to seal an emphatic win for the Swiss league leaders.