The Uruguayan produced a fine performance to claim a 7-6 (8-6) 6-3 victory in one hour and 44 minutes on the clay in Bastad.
Cuevas was seeking revenge over Verdasco having lost to the Spaniard in five sets in the second round of the French Open in May.
World number 111 Cuevas saved two set points in the opener that he took in a tightly contested tie-break.
And he carried the momentum into the second set, racking up 10 aces in total and winning 84 per cent of first serve points as he closed things out to become only the third player from Uruguay to reach the showpiece of an ATP event.
Cuevas will take on Portugal's Joao Sousa in the final after the fifth seed mounted an excellent comeback to beat Argentina's Carlos Berlocq in three sets.
Sousa made sure of a spot in just his second ATP final as he emerged victorious 3-6 6-4 6-4.
Berlocq had won the only previous encounter between the two and looked set to earn a second victory over Sousa when he took the first set in relatively straightforward fashion.
However, despite losing the opening game of the second, Sousa displayed excellent resilience to level the match and force a decider.
And it was Sousa that had the greater firepower in a frenetic third set, breaking Berlocq three times to set up a meeting with Cuevas.