In his first start since joining Valencia from Racing Club in his native Argentina, De Paul provided the decisive moment in the 85th minute, taking advantage of Filipe Augusto's strong work in midfield by thrashing the ball past Vallecano goalkeeper David Cobeno.
The goal put Valencia well on top in the two-legged tie, as they will take two away goals back home for the second leg on December 16.
It could have been much different for the visitors, however, as they trailed 1-0 at half-time following Alex Moreno's goal for Vallecano.
The 21-year-old turned a corner into the net from inside the six-yard box in the 37th minute and the home side could have extended their lead early in the second half but Valencia goalkeeper Yoel Rodriguez Oterino made a brilliant double save to deny Nacho Martinez and Moreno.
The game started to unravel for Vallecano in the 70th minute when Cobeno dropped the ball at Paco Alcacer's feet and the Valencia striker converted the opportunity.
And 15 minutes later, the vistors hit the front in the round-of-32 tie, when Augusto winning the ball just inside Valencia's forward half and then releasing De Paul down the right, with the 20-year-old Argentine striking low and hard across Cobeno into the far corner of the net.
In Cadiz, Villarreal also claimed a first-leg lead and held on for a 2-1 win despite a red card to Adrian Marin.
Gerard Moreno and Nahuel Leiva scored for the La Liga visitors in the first half, while Aitor Arregi hit back for the Segunda B hosts two minutes into the second period.
Marin received a straight red card in the 58th minute but Cadiz could not capitalise on their extra player as Villarreal prevailed.
In the other first-leg clash on Thursday, Real Oviedo drew 0-0 at home to Real Sociedad.
