The 24-year-old Johnson, a two-time NCAA singles champion out of the University of Southern California, dispatched the Spanish veteran 6-3, 6-2 in 1 hour, 13 minutes in their quarterfinal match.
Johnson will tangle with 2012 tournament champ Kevin Anderson of South Africa in the semis. Anderson out-dueled Australian Marinko Matosevic 7-6 (3-7), 6-3, 6-3 in their quarterfinal clash.
Second-seeded John Isner rallied to beat fellow American Rhyne Williams 4-6, 6-3, 6-2 in their quarterfinal match.
Seventh-seeded Marin Cilic of Croatia also advanced with a 6-2, 6-3 win over Teymuraz Gabashvili or Russia. Cilic has won 12 of his last three matches, including the finals in Zagreb two weeks ago.
Rosie O'Donnell unveils nearly 50-pound weight loss
Iranian woman stops the execution of son's killer