Del Potro saved a set point in the tiebreaker before converting his first match point. The 2009 U.S. Open champion from Argentina will play his second final of the season against either Fernando Verdasco of Spain or Milos Raonic of Canada, who played later.
Del Potro dropped only six points on his first serve and hit four aces, while his Uruguayan opponent was unable to convert a single break chance. Del Potro broke Cuevas' serve twice in the first set, but failed on eight break chances in the second as his 65th-ranked opponent took the set to a tiebreaker.
Del Potro, playing his first clay-court event of the season, won in Delray Beach in February in his comeback from wrist surgery last year.
Medina Garrigues of Spain beat Kristina Barrois of Germany 6-1, 6-2 to capture her 10th WTA title.
The 61st-ranked Medina Garrigues was leading 3-2 in the second set when rain interrupted the match.
The break didn't rattle Medina Garrigues, who dropped just seven points on first serve and broke her opponent six times to capture her ninth clay-court event, matching Venus Williams' record among active players.
"Anabel plays with such heavy spin and is such a great clay-court player, it was tough for me on every single point," Barrois said.
The second-seeded Verdasco trailed by a set when play was curtailed by bad light Friday, but he dominated Saturday to pull out a 6-7 (2), 6-2, 6-3 victory. The 12th-ranked Spaniard never faced a break point, while breaking his seventh-seeded South African opponent three times over the final two sets to win.
Raonic's hard-fought 7-6 (4), 4-6, 6-3 win set up the semifinal against Verdasco. The 20-year-old Canadian won both their previous meetings, earning his first title by defeating Verdasco at San Jose.
Raonic, whose quarterfinal match also was carried over to Saturday, overcame a 6-5 first-set deficit to pull ahead against Simon. He rallied from 4-0 down in the second frame to draw even before the Frenchman earned a break point before holding serve to tie it.
Locked at 1-1 in the deciding set, another rain delay forced the players indoors for more than an hour before Raonic broke Simon late in the set to reach his third semifinal of the season.