Rain wreaked havoc on the weekend schedule, forcing the first-round tie to a third day.
Suarez Navarro put Spain up 2-1 on Monday following her second singles victory.
Lucie Safarova returned to the court to keep Czech hopes alive with a 4-6, 6-1, 6-3 win over Soler-Espinosa.
The visitors seized the momentum in the deciding rubber of the best-of-five tie by winning the first-set tiebreaker on the clay.
Germany and Australia meet in the other semifinal, which will be played on April 19-20.